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[pic] 01 INTRODUCTION TO QUANTUM PHYSICS

The fundamental theories of classical physics were developed before the nineteen hundreds. The usefulness of these theories has not diminished with age. However, classical theories do not describe the machinery of the universe at the fundamental levels of atoms, molecules, and elementary particles. This is the province of modern physics or quantum physics.

Quantum theory is the ultimate physical theory that evolved after centuries of scientific endeavor. Many physicists believe that an appropriate combination of quantum physics with relativity will produce a “theory of everything" to describe all phenomena.

Mechanics, electrodynamics, and thermodynamics, are approximations to quantum theory(we say that they are “included” in quantum theory.

Light is a Wave

We know that light is a wave of electric and magnetic fields. One demonstration that shows light is a wave is the double slit experiment. Here light comes through two parallel slits. When the light reaches the screen, it forms a pattern of bars. Waves come from both slits and interfere at the screen; when two crests meet, they form a higher crest; a bright line. When a crest and trough meet, they cancel and produce a dark line. The results clearly demonstrate that light is a wave.

Light is a particle

In order to explain the photoelectric effect, Einstein had to interpret light as little lumps or particles of energy -- photons.

The photoelectric effect works this way: blue light hits a particular metal and electrons fly out. Red light hits metal and no electrons are ejected -even when the red light is intensely bright!

Einstein's explanation was that blue light (high frequency) has very energetic photons that can tear electrons out of the metal. Red light (low frequency) has less energetic photons that are unable to eject electrons.

True or False

For a particular metal, it is possible that green light will eject electrons, but ultraviolet waves will not.

Matter-Wave Duality

De Broglie proposed that light should not be the only thing that exhibits both wave and particle properties. He said all waves have a particle aspect and that all particles have a wave aspect. In short, an object is both a particle and a wave and it reveals one property or the other depending upon the circumstances.

This conjecture was verified by shooting a beam of electrons through a crystal that serves as a kind of “double slit.” A pattern appeared on a screen showing characteristic wave interference.

True or False

a) Light exhibits both wave and particle behaviors.

b) Protons exhibit both wave and particle behaviors

Bohr Atom

Bohr first calculated the energy levels of a hydrogen atom by imposing a non-classical requirement on the orbital angular momentum of the electron. The angular momentum (rp or rmv) was restricted to integer multiples of a basic number called Planck’s constant, (. [rmv = n(] The theory predicted that the atom could realize only discrete orbits and energy levels.

The restriction on orbits may be more readily seen as the result of an electron-wave fitting in standing patterns around the nucleus at just the appropriate distances. Waves may be imagined to look something like a string of sausages so that more sausages (wavelengths) can fit around larger orbits as seen in the diagram.

.

Schrödinger Equation

The de Broglie equation is correct for free particles, but was found to be inadequate for particles affected substantially by interactions. Schrödinger constructed an equation to fully describe the “matter wave” for any system,

[pic].

The Schrödinger Equation is the basic equation of non-relativistic quantum theory. There are two kinds of information given by the equation, eigenvalues and eigenfunctions:

a) eigenvalues

When applied to bound atoms and molecules, the Schrödinger equation predicts that the system has definite, discrete, energies:

El. E2, E:3, ....

No solutions exist for any energies “in-between.” This is in sharp contrast with classical dynamics where there is a continuum of allowed energy values. The equation also determines that other quantities such as angular momentum and spin have similar discrete values. However solutions may also include quantities that are not discrete such as the momentum of a free particle.

The above quantities that are given by the Schrödinger equation are examples of eigenvalues; the only values which the system can exhibit. In general, we look to the Schrödinger equation to provide the appropriate eigenvalues for a specific system.

True or False

(a) Eigenvalues are numbers representing physical properties.

(b) Eigenvalues are always discrete.

(c) Eigenvalues represent all possible characteristics of a system.

(d) Only two energies can be determined from the Schrödinger equation when it is applied to an ammonia molecule in a magnetic field. This means that these two energies and no others are possible for this molecule

b) eigenfunctions

A state is characterized by a set of eigenvalues. For each state of the system, the Schrödinger equation gives a corresponding eigenfunction

[pic].

Each eigenfunction is associated with specific set of observables like energy, angular momentum, and spin. The eigenfunctions belong to a more general category of waves called wave functions that are just combinations of eigenfunctions. We have now to interpret the wavefunctions.

Born Interpretation

Physicist Max Born was first to correctly interpret the wavefunction. Schrödinger and others thought ( was just a measure of the spread of a wavy particle. This belief had flaws that Born pointed out. Suppose that a single electron is shot through double slits. If the pattern that results has bright and dark fringes, then the electron has been divided into pieces.

However, electrons are never seen in parts. It must be that the particle itself is not actually an extended wavy lump. Rather, it is always a point particle when it is detected and wave patterns must be the result of a statistical accumulation of many particles. The wave must be interpreted as the probability that the particle is in a particular position or state. (The larger the wave function at a point, the more likely is the particle to be at that location.)

True or False

Born interpreted eigenfunctions as extended wavy particles.

More on Wavefunctions

When Schrödinger invented the wavefunction, he expected it to be something physical like the density of an extended particle. It turned out not to be a material substance. Rather, it is a ghostly probability wave that dictates the positions and outcomes of physical systems.

( is not the probability itself. The probability of a state is found from the square of the wave function [pic] (much like the energy of an electromagnetic wave is given by the square of its fields).

question

[pic]

The diagram shows the eigenfunction for the location x of an electron in a particular molecule.

(a) Which labeled position is the most likely neighborhood for the electron?

(b) The electron is never at which of the labled points?

Superposition of Waves

An important property of waves is that they “add” (or “subtract”). This is called superposition of waves. Superposition has profound implications for quantum physics

If [pic]and [pic]are eigenfunctions for two possible states, then the system may exist with a wavefunction [pic]that is a combination like [pic]and [pic]. However, the only values that we can obtain from a single measurement are eigenvalues associated with [pic]or [pic], and the probability that the system will be found in condition 1 or 2 depends on the proportion of [pic]and [pic] represented in the combination (superposition).

Example (Schrödinger’s Cat):

A cat spends several minutes in a sound proof box with an atomic device that has a 50% chance to kill the cat. Consequently, the cat is 50% live and 50% dead According to the conventional quantum interpretation, the cat becomes alive or dead (according to chance) when the box is opened for observation.

questions

A photon strikes a glass pane. The eigenfunctions for the photon to be transmitted through or reflected from the pane are (t and (r, respectively.

The wavefunction of the photon is

[pic]

(a) Is this photon more likely to be transmitted or reflected?

(b) In the event that the photon was transmitted, what has its wavefunction become?

Summary of Quantum Theory

Our exposition may obscure the simple fundamentals of quantum physics, so it is desirable to summarize them here as a kind of “how to” list.

a) Given a physical system like an atom or a crystal, write the Schrödinger equation. This is done with a simple set of rules that transforms an energy relation for the system into a differential equation.

b) Solve the Schrödinger equation to find the eigenvalues and eigenfunctions for the system. The eigenvalues are the only physical quantities that the system can exhibit and corresponding eigenfunctions determine the likelihood that a specific state (set of eigenvalues) will be found at a given position.

c) Two specifics determine most applications of the wavefunctions:

(i) systems may be described by wavefunctions that are superpositions of eigenfunctions and (ii) probabilities depend on the square of a wavefunction, [pic]. (The first of these allows alternative possibilities to exist and the second allows the alternatives to interfere.)

Some Philosophical Features of Quantum Physics

Quantum mechanics has a profound influence on the philosophy of nature. Indeed, it has altered our view of objective reality and classical determinism.

In quantum theory, what you know is what you measure (or what some physical system “records”). The acts of measurement and observation can create the resulting state. A system does not have a definite value for a quantity until it is observed. Thus an electron is given a specific spin by an observation; before this, it had only potential spins. A photon in the double slit interference experiment does not pass through a single slit unless we try to detect that slit passage and Schrödinger's cat is not dead or alive until an observation makes it so. Quantum physics does not accept objective reality independent of observation or interaction.

We see that quantum physics differs from classical physics In another basic philosophical sense. Classical physics is deterministic in that when enough initial information is specified, the consequences can be predicted with certainty (the clockwork universe). Quantum theory shows that for a given initial situation, nature can “choose” among alternatives according to predictable probabilities. The theory asserts that nature is indeterministic.

A related deduction in quantum physics is the uncertainty principle that says that a particle’s position and velocity can not both be measured with complete accuracy at the same time. One must sacrifice accuracy in one to increase accuracy in the other. A similar uncertainty exists between the energy and lifetime of a particle.

Another remarkable feature of quantum physics is that the outcome of a process is influenced by other possible outcomes that are not themselves realized.

02 QUANTUM BASICS

Before you study quantum mechanics in a more formal and systematic way, it helps to learn a few basic equations, rules, and examples that are used throughout. Here the topics of the last unit are repeated, but with associated equations and problems.

Light is a Wave

For the present, the most important expression associated with the wave nature of light is the relation between wave velocity v, wavelength ( and frequency f:

|(1) |[pic] |

problem 1.

Find the frequency of light with 4.5 X 10-7 m wavelength. (c = 3.0 X 108 m/s.)

Background Note

Constructive interference of two coherent waves causes the bright fringes in the double slit experiment. This occurs when the two light rays differ only by an integer number n of whole wavelengths:

path difference = n ( constructive interference

Destructive interference causes the dark fringes in the double slit experiment. In this case the two rays differ by odd half-integer numbers of wavelengths:

path difference = (n + ½) ( destructive interference

Light is a particle

Einstein's interpretation of light as a particle depended in part on Planck’s theory of blackbody radiation. Planck was able to describe how the amount of radiation from a heated object depends upon the wavelength and temperature(but only when he assumed that energy was emitted and absorbed in “lumps” proportional to the frequency of the radiation,

|(2) |[pic] |

where the proportionality constant h is a universal constant now called Planck's constant.

|(3) |h = 6.626196 X 10-34 J-s |

Equation (2) was taken by Einstein to be the expression of photon energy, and this is the accepted view today.

problem 2.

Calculate the frequency of a photon emitted when an electron falls from a hydrogen energy level of -3.4 eV to -13.6 eV. (1 eV = 1.60 X 10-19 J)

In the photoelectric effect, the photon energy hf went into the work W needed to rip the electron from the metal, and into the kinetic energy K of the freed electron,

|(4) |[pic] |

W is called the work function and is different for different metals. K in Eq. (4) represents the maximum possible kinetic energy of the electron; as it may be slowed by collisions in the metal.

problem 3.

The work function of Aluminum is 4.25 eV. Calculate the maximum kinetic energy of electrons ejected from Aluminum when irradiated by an ultraviolet beam of wavelength

2.10 X 10-7 m. (1 eV = 1.60 X 10-19 J)

Matter-Wave Duality

De Broglie expressed the wavelength ( of the matter-wave in terms of the momentum of the particle, p = mv:

|(5) |[pic] |

problem 4.

The momentum of a photon is known to be hf/c. Use this fact and the de Broglie relation to find an expression for c in terms of f and (

The following “particle-in-a-box” is a favorite model problem in quantum mechanics.

problem 5.

A particle of mass m is confined in a box of length L such that its de Broglie wave must form a standing wave. This can be done only when half-integer wavelengths fit the box. Use Eq. (5) and the fact that the energy (all kinetic) can be written as

[pic][pic]

and derive the allowed energy levels [pic].

Niels Bohr found the correct energy levels of hydrogen in 1913 by proposing that the electron can only exist in orbits allowed by his “Bohr quantization rule” for angular momentum (:

| |[pic] |

This rule is easily derived from the deBroglie relation.

problem 6.

Derive the Bohr quantization rule for angular momentum by requiring that an integer number n of de Broglie wavelengths fits around the perimeter of a circular orbit [pic].

Schrödinger Equation

The de Broglie relation is accurate for particles that are not subjected to external forces, but it was found to be a special solution of the more general equation of quantum mechanics, the Schrödinger equation.

To construct the Schrödinger equation for 1-dimensional systems, follow these rules:

i) Write the energy relation for the system of interest. The kinetic energy K must be written in terms of the momentum p rather than velocity v,

| |[pic] |

The total energy E including the potential energy V(x) is then given by K(p) + V(x) = E or

|(6) |[pic] |

ii) Replace the momentum p in Eq. (6) by a differential operator [pic]:

| |[pic] |

where [pic] (called “h-bar”) is the Planck constant divided by 2(,

|(7) |[pic] |

Equation (6) is then converted into the operator form

|(8) |[pic] |

iii) Produce the 1-dimensional, time-independent Schrödinger equation by applying both sides of Eq. (8) to the wavefunction ( The result is

|(9) |[pic] |

Schrödinger Wave Equation

The following three problems are exercises in writing the wave equation. It will be important to solve these and other differential equations in later chapters. Here you are given trial solutions to verify.

problem 7.

Follow the rules above to write the Schrödinger equation for a free particle, V = 0. Show by substitution that [pic]is a solution and determine the relation between E and k. (A and k are constants.)

problem 8.

Follow the rules above to write the Schrödinger equation for a particle in a constant potential V = V0, where E > V0. Show by substitution that [pic]is a solution and determine the relation between E, V0, and k. (A and k are constants.)

problem 9.

Write the Schrödinger equation for a simple harmonic oscillator,[pic]. Show by substitution that [pic] is a solution and determine the value of E that accompanies this wavefunction. (A and a are constants.)

Particle in a Box

The particle-in-a-box problem illustrates how the solution of the Schrödinger equation produces a set of energy eigenvalues [pic]and corresponding wavefunctions [pic][pic] This is done by writing the particle's wave equation and finding solutions for ( that satisfy conditions imposed by the probability interpretation.

There is no potential energy inside the box, so the wave equation is the same as in problem b-7. The trail solution [pic] will satisfy the equation, but it must also be made to satisfy the probability conditions.

The probability that a particle is at position x with energy En is proportional to the square of the wavefunction, [pic]. Consequently, if a particle is confined to a box of length L,

[pic],

it must have zero probability of being outside and we require that

|(10) |[pic] |

These are the boundary conditions that must be met by the trial solution.

problem 10.

Write the wave equation for a particle-in-a-box and calculate the energy eigenvalues and wavefunctions. Use a trial solution of the form [pic].

1. AMPLITUDES & PROBABILITIES

The notation and principles of quantum theory are introduced in the next few units. We illustrate these with two simple examples; a one-dimensional particle-in-a-box and a bead on a circular ring.

Dirac Notation and Amplitudes

Dirac notation embraces the quantum philosophy that what you know is what you measure. Consider the one-dimensional particle-in-a-box with allowed energy levels En. The state of this system is completely characterized by the energy level and the Dirac notation for the nth state is |En>.

Writing the symbol |En> infers that we have complete knowledge of the particle's nth energy level, but given this knowledge, what is known about the position x of the particle? The Dirac symbol for what we want to measure is , the probability of it being found in the condition |(> is

|(1) |[pic] ( discrete |

When ( is a continuous (instead of discrete) quantity, [pic]is a probability density and [pic]d( is the probability of a measurement yielding a value in a neighborhood d( of (.

|(2) |[pic] ( continuous |

For the particle-in-a-box amplitude , the probability of a particle with energy En being found in a neighborhood dx around the point x is ||2dx. Amplitudes of the form that are probability densities for position and are called wave functions. Not all amplitudes are wave functions; and are counterexamples.

problem 2.

Use probabilities or probability densities where appropriate and identify any amplitudes that are wave functions:

(a) Write the Dirac notation representing the probability for a particle with energy En to have momentum p.

(b) Write the Dirac notation representing the probability for a particle with energy E2 to be found with different energy E1.

(c) A bead on a ring can be completely specified by its momentum p. Write the Dirac notation representing the probability for the bead with momentum p to have position x.

The amplitude for position (a wave function) must be a continuous, and single-valued function of position x.

problem 3.

Given that the wave function for a bead on a ring of circumference L is

[pic],

find (a) the probability of an n=-3 particle being between x=0 and x=L/4 and

(b) the probability of particle with arbitrary n being in the interval between x=0 and x=L.

problem 4.

Given that the wave functions for a particle in a box of length L are

[pic],

find (a) the probability of an n=2 particle being in the “middle” between x=L/4 and x=3L/4 and (b) the probability of an n=1 particle being in the same interval. Sketch the probabilities for both cases and comment on the differences. ans:1/2, 1/2 + 1/(

2. VECTORS & SUPERPOSITION

Kets describe that which is known about systems and the mathematics obeyed by kets is that of vectors. This unit describes the vector properties and their physical consequences. For the present, we continue to treat amplitudes like as “given” information instead of deriving them.

Review of Vectors

A linear vector space is a set of objects called vectors a, b, c,... for which the following operations are defined:

1. Addition: the sum of two vectors is a vector and addition is commutative and associative

[pic]

2. Multiplication by a Scalar: multiplication by any complex number is distributive and associative.

[pic]

3. A null vector 0 exists in the space such that

[pic]

4. Every vector a has an additive inverse -a such that

[pic]

problem 1.

Show that the following are vector spaces: (a) displacements in 2-D space and

(b) the set of all continuous functions f(x) defined on the interval (0,L).

Base Vectors

Any two-dimensional vector v can be written as a sum of the form v = vxi + vyj. Generally, when any vector in a vector space can be expanded similarly with a minimum of N vectors e1, e2, ..., eN,

|(1) |[pic], |

the space is said to be N-dimensional. The c's are called components and the e's are called base vectors. When Eq.(1) is satisfied for all vectors in the space, we say the base vectors are complete. A set of base vectors is a basis of the space.

problem 2.

A vector in 2-dimensional Euclidean space has a magnitude of 10 units and is inclined at 60o above the x-axis. (a) Expand this vector in unit vectors i and j. (b) Write this vector in column matrix form and expand it in column vectors representing i and j.

The Fourier theorem shows that a function f(x) defined on the interval (0,L) can be expanded in an infinite series of sine and cosine functions. In general,

|(2) |[pic] |

This can be viewed as a special case of Eq.(1).

problem 3.

Identify the base vectors in Eq.(2). What is the dimension of the function space?

Scalar Products

A scalar product of two vectors |a>, |b> is written and has the following properties:

1. = *

2- = +

3. > 0 and = 0 implies |a>=0

Two vectors whose scalar product is zero are said to be orthogonal. The length of vector |a> is defined as ½

problem 4.

Write definitions of the specific scalar products for the following: (a) Euclidean vectors, (b) column vectors, (c) functions f(x). In each case, demonstrate that the scalar product properties are satisfied.

problem 5.

Calculate for the following: (a) u=5i cos60-5j sin60 and v=10i (b) u= (4, -1, -5) and v= (2, -3, 2) (c) u= (1+i, 1-2i) and v= (2, -i) (d) [pic] and [pic] on the interval (0,L) (e) [pic] and [pic] on the interval (0,L)

problem 6.

Calculate the lengths of the u vectors in the previous problem.

problem 7.

Show that u=5i cos60-5j sin60 is orthogonal to v=i cos30+j sin30. Which of the vector pairs in Prob.2 are orthogonal?

Unit Vectors

Ordinary Euclidean vectors are easily treated when they are expanded in terms of unit vectors i, j, k. Similar useful expansions are also available for more general vectors—including functions.

Notice that i, j, k are orthogonal and have unit lengths. In the scalar product notation; =0 and =1, etc. In order to have base vectors that are as convenient as these, it is usual to use a basis that is both orthogonal,

|(3) |[pic] |

and normalized to a unit length,

|(4) |[pic]. |

When the base vectors satisfy both Eqs.(3) and (4), we say the basis is orthonormal.

problem 8.

Show that i, j, k are orthonormal.

problem 9.

Show that the set of functions [pic] with n=0, 1, 2,... are orthonormal.

problem 10.

Show that the set of functions [pic] with n = 1, 2,... are orthonormal.

Linear Vector Space Principle

Physical states |(> are vectors in a linear vector space and amplitudes [pic] are associated scalar products.

This principle has profound consequences. First, we note from the definition of the scalar product that

|(5) | = of a system can be expanded in terms of any base states |u1>, |u2>,... Thus the state of polarization of a photon is a sum of left- and right-polarized photons and the vitality of Schrödinger’s cat is a sum over the live and dead cat states. This feature is called the superposition principle. It can be written symbolically as

|(6) |[pic] |

where we will assume the base states are orthonormal.

problem 12.

A special case of Eq.(6) is |(> = 0.8|u1> + 0.6|u2>.Show that the u1 component equals and the u2 component equals . 2-14 Show that Eq.(6) can be written as

|(7) |[pic] |

problem 13.

Write the state |(> of Schrödinger’s cat given that the amplitude for it to be alive is [pic]. Use the symbols |L> and |D> for the live and dead states.

The most succinct expression of the superposition principle follows from Eq.(7),

|(8) |[pic] |

problem 14.

Use the orthonormal functions [pic] and Eq.(8) to expand an arbitrary function f(x)= in terms of these functions and coefficients . The result is a Fourier series.

summary:

Quantum states are vectors. As a consequence, an arbitrary state |(> can be expanded in base vectors |un> that represent a complete set of known states. It is then said that |(> is a superposition of |un> states. See Eqs.(6) and (7) for expressions of superposition.

Averages

As an example of averaging with probabilities, consider averaging quiz grades for an imaginary class of 10 students. The possible grades are 0, 50, and 100. One student got 0, four got 50, and five got 100. Here is the class average:

[pic]

Another way to look at the same average is:

[pic].

Notice that the probability of a student getting 0 is 1/10 and the probability of getting 50 is 4/10, etc. We write

[pic].

This is a special case of a general way to take an average over g1, g2,( when the corresponding probabilities p1, p2, (are known:

[pic]

In quantum mechanics we often have a continuous range of possible outcomes (think 0 to 100 with any fraction between). Then we modify the average above using a probability density (:

[pic]

In this case, average g is ,

[pic]

We expect ( in quantum mechanics to be (*(. A slight wrinkle occurs in quantum theory where we make a “sandwich” of the thing being averaged so that it looks like this:

[pic]

This form works even when g represents an operator like momentum ([pic]).

3. OPERATORS & OBSERVABLES

Objects that transform vectors according to some rule are called operators. In quantum physics, operators represent the instruments or objects that measure or otherwise record physical quantities. An operator ( (measuring device) is applied to a state ( (system in a specific condition) in order to render a value of the observable (.

|(1) |[pic] |

For example, the momentum operator P is applied to the bead-on-a-wire wavefunction in order to measure the momentum of the corresponding system.

In quantum physics, the acts of measurement and observation create the resulting state. A system does not have a definite value of a quantity until it is observed. Thus a bead-on-a-wire is given a specific momentum by an observation; before this, it had only potential momenta. Schrödinger’s cat is not dead or alive until an observation makes it so.

Linear Operators

A linear operator ( transforms a vector |a> into another vector (|a> according to a rule that obeys the following linear relation:

|(2) |( (c1|a> + c2|b>) = c1(|a> + c2(|b> |

Operators may also operate “backward” on a bra-vector

|(3) |when (|a> = c|b> then or Vo). The probability of reflection will depend on the initial energy E and the barrier “height” Vo

The relative probability that the particle will be reflected is the ratio R called the reflection coefficient,

R = |B exp(-ikx)|2 / |A exp(ikx)|2

or

|(11) |[pic] |

Since the particle must be either reflected or transmitted, the sum of R and the relative probability for transmission T, the transmission coefficient, must add to unity:

R + T = 1 .

Equation (11) requires a relation between A and B in order to evaluate the ratio R. This is established from the boundary conditions for a smooth connection at the boundary; [pic] and [pic].

problem 14.

Calculate reflection and transmission coefficients for a particle incident on a potential step Vo < E.

4. HARMONIC OSCILLATOR

Here we analyze a system of great importance in physics, the quantum mechanical harmonic oscillator. The physics and mathematics of the oscillator are relevant to the emission and absorption of light by matter (blackbody radiation), the analyses of radiation and fields, the treatment of systems of identical particles, and other basic problems.

The methods used in solving the harmonic oscillator problem are illustrative of the standard techniques for treating other bound systems. These include i) wave mechanics, where an eigenvalue problem is represented as a differential equation (usually the Schrödinger equation), ii) matrix mechanics, where an eigenvalue problem is represented with matrices applied to column vectors, and iii) abstract operator mechanics, where an eigenvalue problem is solved using abstract symbols for operators and states. The last approach is the most sophisticated, but we will emphasize it because, with some modifications, it can be used to solve basic angular momentum problems and the hydrogen atom relatively easily.

The Classical Oscillator

The kinetic energy of a particle of mass ( and momentum p is p2/2( and the potential energy of an oscillator can be written as ((2x2/2, where ( is related to the spring constant k by the relation

|(1) |[pic] |

The classical Hamiltonian is the sum of kinetic and potential energies,

|(2) |[pic] |

and conservation of energy E can be expressed as

[pic]

The quantum analogue of this expression is an eigenvalue problem for energy:

|(3) |[pic], |

where H is an operator having the same form as Eq.(2), but with scalars p and x replaced by operators [pic]and[pic].

Wave Mechanics for Oscillator

When the state in Eq.(3) is taken to be in the x-representation, , the expression becomes a differential equation called the (time-independent) Schrödinger equation:

|(4) |[pic] |

problem 1.

Construct Eq.(4) from Eqs.(2) using the momentum operator given in Unit 3.

Equation (4) is the time-independent Schrödinger equation for the harmonic oscillator. Its solution is shown in all quantum mechanics texts. Bound state solutions exist only when

|(5) |[pic] |

where n is an integer. Each value of En has an associated wavefunction (n = (this is written more simply as ), and some solutions are graphed below.

n = 1

n = 3

n = 10

classical case corresponds to n((

Abstract Operator Approach (Dirac)--Eigenvalues

The eigenvalues of the Harmonic oscillator can be found from the following information alone, without the use of the Schrödinger equation:

|(6) |[pic] |

|(7) |[pic] |

where H is given by Eq.(2). The core idea is to construct “raising and lowering” operators a+ and a with the ability to produce one state from another:

a+|n> = constant |n+1>

a|n> = constant |n-1>

These are then applied repeatedly to Eq.(7) with the result that all the possible energy levels are produced.

In the following problem sequence, let [pic]. These quantities can be restored in the results by dimensional analysis. (Since we are dealing with three fundamental quantities—mass, time, and distance—three units may be arbitrarily chosen to specify the values of three quantities.)

problem 2

Define the operator [pic] and derive the relation

|(8) |[a, a+] = 1 |

problem 3

Show that H = = a+a + ½.:

|(9) |H = = a+a + ½. |

problem 4

Demonstrate that a+a is the “number operator,” that is,

|(10) |a+a |n> = n |n>. |

problem 5

Use the result of the last problem to show the eigenvalue spectrum of the Harmonic oscillator is given by Eq.(5). (Use dimensional arguments to restore [pic].)

Eigenfunctions in the Dirac Approach

Eigenfunctions can be generated by first finding the lowest eigenfunction and then “raising” it repeatedly to , , etc.

problem 6

Use the fact that results in an eigenvalue for each:

A|(> = a|(>

B|(> = b|(>.

Premultiply the first by B and the second by A. Subtracting gives (AB-BA)|(> = 0, and the result [A,B]=0 follows. If the eigenstates were different at the outset, we could not get this result.

5-1 Show that if any two observables can be measured simultaneously, their operators must commute.

5-2 For the following, indicate whether the operators in brackets can or cannot have simultaneous values: (a) [p, p2/2m], (b) [x, y]=0, (c) [x, p]=i(, (d) [Jx, Jy]= i(Jz (this defines angular momentum), (e) [Jz, J2]=0

States and Probabilities in 2-D

A particle's position in two dimensions can be specified by two values, x and y. The position state can therefore be symbolized by |x, y>. Similarly, momentum has two components in two dimensions and a momentum state can be indicated as |px ,py>. The wavefunction for the latter state is written as .

These examples illustrate two general requirements. First, the quantities in a bra or ket must be able to be specified simultaneously. This means that their operators must commute. The examples above originate with the relations [x,y]=0 and [px,py]=0. Secondly, the number of observables specifying a state must be equal to or greater than the number of dimensions. (In most practical cases these entries are equal in number to the number of dimensions.) In two dimensions we need two eigenvalue entries like |x,y> or |px,py>.

The requirements can be summarized by saying we must have a complete set of commuting observables for a given system. This means that for an n-dimensional system we must specify at least n operators that are linearly independent and mutually commuting. Consider, for example, a two-dimensional harmonic oscillator with the following commutation relations: [x,H](0, [px,H](0, [J,H]=0. The first two expressions show that there cannot be states like |x,E> or |px,E>. The last expression shows that angular momentum and total energy commute so J and H are a complete commuting set and |j,E> is a legitimate state.

5-3 Consider a free particle in space. Using the information in the following commutation relations, select a convenient set of complete commuting observables and write a corresponding ket label:

[x,px]=[y,py]=[z,pz]=i(,

[px, py]=[py, pz]=[px, pz]=0.

5-4 A particle is confined to the surface of a sphere. Using the information in the following commutation relations, select a convenient set of complete commuting observables and write a corresponding ket label:

[Jx, Jy]=i( Jz

[Jy, Jz]=i( Jx

[Jz, Jx]=i( Jy

[Jx, J2]=[Jy, J2]=[Jz, J2]=0

(Note that in the last problem the linear momenta are related because the particle is constrained to move on a sphere. This causes the relations given in problem 5-4, [px,py]=[py,pz]=[px,pz]=0, not to hold in this case.)

5-5 An electron is bound to a stationary proton (Hydrogen atom). Using the information in the following commutation relations, select a convenient set of complete commuting observables and write a corresponding ket label:

[Jx, Jy]=i( Jz

[Jy, Jz]=i( Jx

[Jz, Jx]=i( Jy

[Jx, J2]=[Jy, J2]=[Jz, J2]=0

[Jx, H]=[Jy, H]=[Jz, H]=0

[J2, H]=0

Probabilities in Two Dimensions

Calculating probabilities from amplitudes in higher dimensions is practically the same as for one dimension. The quantity P(m,En|() = | to be in a discrete state with angular momentum m and energy En.

Probability densities are only slightly more involved. The quantity P(x,y|() = | to be in a neighborhood dx dy of point (x,y). In three dimensions, the neighborhood is dx dy dz. If two particles are involved with labels like (x1,y1,z1,x2,y2,z2), then the "volume element" to be integrated over is dx1dy1dz1dx2dy2dz2. If the bra is labeled by, say, two continuous momenta, are expressed in wavefunction form and the energy eigenvalue problem is

|[pic][pic] |(2a) |

Schrödinger’s equation for hydrogen results when this is converted to wave-mechanical form with differential operators and with kets replaced by wave functions. Usually, textbooks find the allowed eigenvalues and eigenfunctions by solving the Schrödinger differential equation. It is a tedious process and here we will find solutions by an easier operator approach closely akin to our harmonic oscillator solution.

Equation (2a) can be simplified by eliminating the [pic] operator. Since [pic], we can write

|[pic] |(2b) |

where

|[pic] | |

| |(3) |

In the following sections, we solve the eigenvalue problem (2b) by rewriting [pic]in a “factored” form [pic]. The [pic] operator will raise the [pic]eigenvalue and [pic]will lower it.

H in Factored Form

Define an operator [pic] with the form

|[pic] |(4) |

where

|[pic] |(5) |

The problem sequence that follows will determine [pic]in terms of the A+ and A operators.

6-1 Show that [pic]

6-2 Given the required commutation relation between r and its conjugate momentum, [pic], show that [pic].

3. Use the results of problems 6-1 and 6-2 to derive the form

|[pic] |(6) |

.

Raising and Lowering Properties

The next two problems determine the raising and lowering properties of the A operators.

6-4 Show that [pic]. (Use the results of problems 6-1 and 6-2.)

6-5 Demonstrate that [pic]is proportional to [pic]; that is, [pic] is a raising operator for [pic]. Similarly, show that [pic] is proportional to [pic]such that [pic] is a lowering operator for[pic]. Note that these operations do not change the value of E.

Eigenvalues and Eigenfunctions of Hydrogen

The energy of the hydrogen atom must dictate upper and lower bounds for angular momentum, so the A operators cannot raise or lower without limit. Let [pic]be the maximum angular momentum at a particular energy E so that [pic]cannot be “promoted” to higher [pic]:

|[pic] |(7) |

This condition enables us to determine eigenvalues & eigenfunctions of hydrogen.

6-6 Show that the energy eigenvalues for operator [pic] in Eq. (6) are given by

|[pic] | |

| |(8) |

where n = 1, 2, 3,...

Hydrogen eigenfunctions can be written in separated form,

|[pic] |(9) |

where Rn(r) is the radial wavefunction. It is easily determined when operator pr is expressed in differential operator form:

|[pic] |(10) |

(This expression can be verified from the commutation relation [pic].)

6-7 Find the hydrogen radial eigenfunctions (un-normalized) for a maximum allowed value of angular momentum. (Use Eqs. (7) and (10).)

6-8 Calculate the radial ground state R10 of the hydrogen atom. Demonstrate that the probability density[pic] is a maximum at the Bohr orbit. (Use [pic])

6-9 Find the normalized radial wavefunction for hydrogen R20.

Selection Rules for Hydrogen

Transitions between Hydrogen energy levels can only occur when the change in total angular momentum changes by 1:

[pic]

As a corollary, this requires [pic].

Background

Classical electrodynamics says an accelerating charge e radiates power P according to

[pic]

where a is the acceleration and [pic].

problem

Use dimensional analysis to find [pic].

Quantum Requirement for Radiation

The acceleration of an oscillating electron is given in three dimensions by a = –ω2 r , so average power can be written as

[pic].

The quantity er is the (average) dipole moment of the atom. Only non-vanishing averages [pic] allow transitions between the primed and unprimed levels. It can be shown that the integrations over θ vanish unless [pic].

problem

Show that [pic] and [pic] using only integrations over θ.

7. MATRIX MECHANICS

We saw that matrices are one kind of operator and column vectors are the associated kind of state. Here we will show that all linear operators have a corresponding matrix and all ket vectors have a corresponding column vector. As a consequence, all quantum mechanical problems can, in principle, be cast into matrix and column vector form. This formulation is called matrix mechanics to distinguish it from wave mechanics where differential operators and wavefunctions are used. In practice, both forms of quantum mechanics are used side by side along with the abstract Dirac approach. Usually the form of quantum mechanics chosen is the one that makes the current problem easiest.

Matrices Column Vectors

A matrix associated with an operator ( is an ordered array of numbers (i,j given by the expression

|[pic], |(1) |

where i and j represent sets of quantum numbers specifying the states. Note that the specific form of the matrix depends on the base states used to describe it. We say the description in a particular basis is a representation of the operator.

7-1. Spin ½ corresponds to angular momentum (J2=S2, Jz=Sz) with j = ½ and m = –½ ,+½ . (a) Use the definition (1) to derive a matrix for Sz . (b) Find the eigenvectors and show that

[pic] and [pic]

are the respective raising and lowering operators and use these to construct Sx and Sy.

7-2. Evaluate the matrix elements x0,0 and x0,1 for the simple harmonic oscillator. Indicate an infinite matrix for the Hamiltonian of the harmonic oscillator (Hi,j).

Column Vectors

A general vector |Ψ> has a column vector form given by the array

|[pic] | |

| | |

| |(2) |

Again, note that the column vector depends on a specific representation.

7-3. Calculate normalized column eigenvectors for spin Sz directly from the eigenvalue equations. Show that the matrix for Sz gives the appropriate eigenvalues when it is applied to these vectors.

7-4. Given a state of Schrödinger’s cat as [pic], calculate the corresponding eigenvector in the “live” and “dead “basis.”

8. TIME DEVELOPMENT

Most often, it is convenient and desirable to use eigenstates |ω> that correspond to conserved quantities of operator Ω. Since conserved quantities do not change with time, we have not yet had occasion to see any explicit time development. Here we give the time development principle.

Time Development Principle

The Hamiltonian operator [pic]determines the time development of a quantum mechanical system:

|[pic] |(1) |

or

|[pic]. |(1’) |

8-1. Show that a formal solution to Eq.(1’) is

|Ψ(t)> = |Ψ(0)> exp(-itH/()

8-2. Show that when observable Ω commutes with H, [Ω,H] = 0, then Ω is conserved.

8-3. Write the time-dependent Schrödinger equation for a simple harmonic oscillator.

APPENDICES

A6 HYDROGEN EIGENFUNCTIONS

The following is a list of some normalized wavefunctions for hydrogen [pic] for various values of n, , and m. a0 denotes the Bohr radius.

|[pic]. | |

The separated form for the wavefunctions is

|[pic] |(1) |

where

|[pic] | |

| |(2) |

[pic]

A7 Identical Particles

The Eigenvalues of Exchange 1

Fermi and Bose Particles 2

Fermi Particles 2

Bose Particles 2

Quantum Statistics 3

Applications 4

Free Electron Model for Metals 4

Density of States for Electrons 5

Black-Body Radiation 6

However similar they are, the members of a set of twins or a pair of billiard balls are clearly different objects. Fundamental particles, however, are identical in all respects so that when two electrons in the same state are interchanged, there is no possible way to distinguish the exchanged system from the original.

The simple fact that the interchange of identical particles cannot change a system has profound consequences. We will find that only two kinds of particles can exist. For one kind, bosons, an unlimited number can coexist in the same state (including the same place). The other kind, fermions, can have only one particle per state.

The Eigenvalues of Exchange

Consider a system with two identical particles at positions [pic]. The system is governed by a wavefunction [pic]. We can exchange the particles so that the positions are reversed producing a wavefunction [pic]. This is expressed mathematically by having an exchange operator applied to the original wavefunction,

|[pic]. |(1) |

Apply to both sides of Eq. (1) to produce

|[pic]. |(2) |

If there is a physical consequence of indistinguishability upon exchange, there must be an eigenvalue problem of the form [pic]where is some observable. Clearly, Eq. (2) must correspond to [pic] and we find that

|[pic] |(3) |

Particles that exhibit the + sign upon exchange so that [pic] are called bosons where those that exhibit the - sign with [pic]are called fermions.

problem

Derive [pic] starting from Eq. (1).

Fermi and Bose Particles

When two particles are not interacting, their composite wavefunction is a product of the independent wavefunctions [pic], where a and b refer to different quantum numbers. To see this, write

|[pic]. |(4) |

Setting

|[pic], |(5) |

we can separate Eq. (4) into two parts,

|[pic] |(6) |

where

[pic]

problem

Derive Eqs. (6) from Eq. (4).

Fermi Particles

Notice that the product function [pic] does not obey [pic] for either eigenvalue [pic]. However, it is easy to check that the exchanged product [pic] is also a solution to Eq. (4). The difference of both products is a solution that satisfies the Fermi condition,

|[pic] |(7) |

with

|[pic] |(8) |

problem

Show that Eq. (7) is satisfied by expression (8).

Now note that Ψ’0 when a=b. That is, there is no chance that two identical fermions can coexist in the same quantum state! This stuff is familiar matter. It includes electrons, protons, and neutrons—in fact, all particles with half-integer spins, [pic],are Fermi particles.

Bose Particles

The eigenstate with [pic],

|[pic], |(9) |

applies to bosons and is easily constructed from the sum of the two products:

|[pic]. |(10) |

problem

Verify that expression. (10) satisfies [pic].

You can see immediately that [pic]does not vanish when a=b. An unlimited number of these particles can occupy the same state. These particles are usually associated with fields and the most familiar examples are photons and mesons. Bosons all have integer spins, 0, 1, 2,..

problem

Identify the following particles as fermions or bosons: [pic]

Quantum Statistics

The indistinguishability of particles has consequences for the macroscopic behavior of particle aggregates. For instance, we can describe conductivity with a free electron model and blackbody radiation with a photon gas model. The passage from quantum mechanical information to macroscopic physics is effected by the methods of statistical mechanics. Here we introduce some results of statistical mechanics without proof:

Individual particles with energy εj have an average occupation number nj given by the following expressions:

|[pic] | |

| | |

| | |

| |(11) |

(The Maxwell-Boltzmann distribution applies to distinguishable particles)

Applications

There are many applications of the distributions above. Here we focus on the free electron model and the Planck formula for blackbody radiation.

Free Electron Model for Metals

[pic]

When we plot the Fermi-Dirac distribution for very low temperatures, we get the curves shown in the diagram. The curve drops to half the maximum value at [pic] where [pic] is the value of [pic] at T=0. This energy is termed the Fermi energy and is roughly the typical energy per particle near zero temperature. In metals, room temperature is a good approximation to .T=0.

In order to calculate [pic] for the free-electron model, we need to convert to a continuous spectrum for . That is,

|[pic] |(12) |

where [pic] is the number of states in energy interval d. is called the density of states factor. To proceed with the calculation of N, we need to determine .

Density of States for Electrons

The eigenvalues of momentum for a particle in a cube of length L are

|[pic] |(13) |

with similar expressions for the y and z dimensions. The n’s can be approximated as continuous variables in a macroscopic volume so an increment of the total number of states is

[pic]

We find

[pic].

The product [pic] is a volume element in momentum space in direct analogy with the standard spatial volume. This can be converted to spherical coordinates using [pic]=[pic] where the total momentum magnitude p is the analog of radius r. Finally, we must (i) divide by 8 because only positive values of n’s (and p’s) are used and this applies to only [pic] of a sphere of radius p; (ii) multiply by 2 to account for the two possible spin states of an electron. The result is

|[pic] |(14) |

problem

Derive Eq. (14).

We want to identify the left hand side of  (14) with [pic] so we need to convert from momentum to energy. Since

[pic],

Eq. (14) becomes

|[pic] |(15) |

Now we can complete the calculation of Eq. (12). Note from the diagram that at T=0, n=1 below [pic] and n=0 above [pic]. The integral can be solved for :

|[pic] |(16) |

problem

Complete the integral in Eq. (12).

A numerical estimate for a typical metallic lattice spacing of 3X 10-10 m gives

[pic].

so that

[pic].

This Fermi energy is huge compared with typical thermal energy [pic] That is, room temperature is a lot like absolute zero to conduction electrons.

We see that the cathode of a vacuum tube must be heated to increase electron emission. Likewise, a photomultiplier must be cooled to decrease spontaneous emission of electrons. Calculations similar to those done here can establish the heat capacity of the metal and detail the features of thermionic emission.

Black-Body Radiation

Light and radiation in equilibrium with heated material can be treated as a photon gas. The model material is termed a “black body” because it absorbs all the radiation incident upon it. The energy of radiation in volume V is given by the expression

|[pic] |(17) |

where n is the Bose-Einstein distribution (with =0 because photons have no ‘rest’ energy) and is the energy density of states for photons.

Fortunately, we have done most of the work already. Photons have two polarization states (corresponding to two spin states for electrons) and Eq. (14) applies to the number of photon states. Photon momentum and energy are given in terms of frequency ,

|[pic] |(18) |

so that Eq. (14) becomes

|[pic] |(19) |

The integral of Eq. (17) can now be assembled:

|[pic]. |(20) |

The integrand of this expression enables us to write the energy density per unit frequency interval (),

|[pic] |(21) |

This is the famous Planck formula.

problem

Derive the Planck formula from Eq. (17).

Integrating Eq. (20) is not trivial. It can be accomplished using

[pic]

giving the detailed T4 law,

|[pic] |(22) |

A8 Dirac Delta Function

Delta Function Concept 1

Kronecker Delta 1

Dirac Delta 1

Properties and Representations 2

Properties 2

Representations of the Delta Function 3

Fourier Integral Theorem 3

Other Representations 3

Applications 4

Boundary Conditions 4

Bound State 5

Reflection and Transmission 5

The Kronecker delta [pic] is familiar from relations like [pic]. In this case the states are discrete like the eigenfunctions of a one-dimensional well or harmonic oscillator. However, quantum states may also be continuous like the eigenstate of free-particle momentum, [pic]. The Dirac delta function [pic] is the analog of [pic] for inner products of continuous states, [pic].

Delta Function Concept

We introduce [pic]by first writing properties of [pic] for discrete sums. We then insist that the change [pic] preserves these properties when we convert the sums to continuous integrals, [pic].

Kronecker Delta

The Kronecker Delta is defined by

|[pic] |(1) |

and two central properties are

|[pic] | |

| |(2) |

problem

Evaluate the sum [pic] by expanding it and using the definition (1).

Dirac Delta

Changing from sums in Eqs. (2) to integrals and changing the notation by the replacement [pic], we write

|[pic] |(3) |

[pic]

where the integration intervals include the neighborhood around x=0. These properties can be realized by a curve like the one shown in the diagram. It is zero everywhere except for a narrow spike of unit area around x=0.

Imagine now that the rectangle is made narrower and taller so that the area remains the same but the width approaches zero and the height approaches infinity. The resulting Dirac delta function satisfies the desired conditions and is defined by the relations (3) and

|[pic] |(4) |

problem

Evaluate (a) [pic], (b) [pic].

Properties and Representations

Properties

Several delta function properties are collected here: Among the most important is

|[pic] |(5) |

problem

Prove Eq. (5). Hint: change the variable from y to x = y-a then use one of the basic relations (3)

problem

Evaluate [pic]

problem

Prove the following properties.

|[pic] |(6) |

Representations of the Delta Function

The delta function is written in many ways. The most important of these is based on the Fourier integral theorem.

Fourier Integral Theorem

|[pic] | |

| |(7) |

A way to look at this theorem is to suppose you are given a function f(x) that you want to expand in the base states exp(ikx). The second expression shows how to find the coefficients of the expansion and the first is the expansion. You can use this view to prove the most important delta function representation:

|[pic] |(8) |

problem

Prove Eq. (8).

Other Representations

[pic][pic]

Applications

Delta functions are easy to manipulate and are used here to analyze simplified versions of standard problems. We treat a bound state of a delta function well and reflection-transmission from a delta function barrier.

Boundary Conditions

The boundary conditions imposed on the usual square well or square barrier are that at the interface of regions I and II obey

|[pic] | |

with similar conditions at the interface of regions II and III.

[pic]

When the barrier or well is infinitesimally narrow, the wavefunction must still match on both sides, [pic], but the slope changes abruptly. This can be seen by writing the Schrödinger equation with potential [pic] and integrating over an infinitesimal interval [pic] surrounding x=0. Taking the limit [pic] gives the result

|[pic] |(9) |

problem

Show that in the neighborhood x=0 of a potential [pic], the boundary condition for slopes is given by [pic].

Bound State

The bound state energy of delta function well [pic] is calculated as follows: (a) Find solutions to the Schrödinger equation for regions outside of x=0 that assure [pic]. The form for the wavefunction to the left and right of the potential are then [pic] where is a positive real constant, (b) Apply the boundary conditions at x=0:

[pic]

This imposes a condition on the constant that determines the allowed energy.

problem

Find the bound state energy and wavefunction for [pic]. Sketch the wavefunction and well. [pic]

Reflection and Transmission

Reflection and transmission from a delta function barrier [pic] is calculated as follows: (a) Find a solution to the Schrödinger equation for the region to the left of x=0 that describes waves traveling both left and right (reflected and incident waves). The form for the wavefunction here is [pic] where k is a positive real constant. (b) For the region to the right of the delta function, find a solution to the Schrödinger equation that represents a transmitted wave moving to the right, [pic]. (c) Apply the boundary conditions at x=0:

[pic]

These give relations between the coefficients. (d) Determine the reflection and transmission coefficients using any two of [pic], [pic], and R + T=1.

problem

Find reflection and transmission coefficients for the potential energy [pic]. At what incident energy are reflection and transmission equally likely?

A9 Perturbation Theory

Basic Problem 1

First Order Results 1

Derivation 2

Applications 4

Alkali Atoms 4

Normal Zeeman Effect 5

We can find exact solutions to only a few basic systems like the simple harmonic oscillator and the Hydrogen atom. More complex systems must be treated with approximation techniques. Foremost among these is the perturbation approach where we neglect relatively small terms to arrive at a simple system (however artificial) that can be solved. Then we add back corrections (perturbations) due to the formerly neglected quantities.

As an example, consider finding the ground state energy of He. If we neglect the interactions between the electrons and consider only the forces between the nucleus and each electron, the wavefunction is made of Hydrogen wavefunctions. The electron—electron interaction is then treated as a perturbation and its approximate contribution to energy is added to the energy of the simplified system.

Basic Problem

The basic time-independent perturbation problem is as follows: Consider a Hamiltonian H,

|[pic], | |

where H0 is the unperturbed Hamiltonian for a fully solved system and V is a relatively small perturbation term. Find corrections to the unperturbed eigenvalues and eigenfunctions to various degrees of accuracy.

First Order Results

At present, we limit ourselves to the simplest case of first order corrections. Denote the energy and eigenstate of the unperturbed system as E0 and [pic] (here the zero does not necessarily refer to the ground state). Then we have

|[pic] |(1) |

and the correction to the energy (to first order) is

|[pic] . |(2) |

The first order eigenstate is given by the expression

|[pic] |(3) |

where the sum is over the possibly infinite number of unperturbed eigenstates. In practice, only a few states near the original [pic]are included.

problem

Find first order corrections to both energies and wavefunctions for a particle in a rigid one-dimensional box of length L containing potential [pic] where is a small parameter.

Derivation

A clever but simple derivation produces Eqs. (2) and (3). Introduce an artificial parameter as a coefficient for the presumed small interaction V so that

|[pic]. |(4) |

Now we can think of the solutions as power series in and write

|[pic], | |

| |(5) |

where the superscripts indicate the order of the correction. Substitute these into the exact expression,

|[pic] |(6) |

and since Eq. (6) must hold for an arbitrary value of , the coefficients of each power ofmust be independent and equal on each side of (6). We obtain

|[pic] | |

| |(7) |

problem

Derive the intermediate results (7).

The first order approximations can be obtained from the second of Eqs.  (7), but first the unknown kets [pic] must be re expressed in terms of known quantities; in particular, they are expanded in terms of the complete base states of the unperturbed problem [pic]:

|[pic]. |(8) |

Substitute and apply the bra [pic] to both sides of the expression so that the normalization condition [pic] gives Eq. (2). The coefficients an are similarly determined by applying the bra [pic] and using the orthonormality condition [pic]. Equation (8) then gives the first order state of Eq. (3).

problem

Derive Eqs.  (2) and (3) from Eqs.(7).

Applications

Perturbation theory is applied so widely that almost every branch of modern physics will provide examples. Here we give two relatively simple physical applications of first order perturbation theory.

Alkali Atoms

The alkali elements have one outer electron orbiting an inner core of electrons. On average, this inner core can be treated as spherically symmetric. When the outer electron is far from the core, it sees only a single charge and has a Hydrogenic energy level. When it penetrates the inner core, it sees a larger positive charge and is perturbed from the Hydrogen level.

[pic]

We can consider the outer electron energy to be a perturbed Hydrogen level. The situation is shown in the diagram where the total interaction experienced by the valence electron is the sum of a single proton and a combined point charge +(Z-1)e surrounded by a uniform cloud of negative charge -(Z-1)e

problem

Show that the potential corresponding to the above diagram is given by

[pic]

[pic]

The degree to which the outer electron penetrates the inner core depends on its angular momentum . This agrees with classical ideas where the lower the angular momentum, the closer the orbit approaches the nucleus. It follows that [pic] is larger for smaller .

Note that the perturbation V becomes increasingly negative for decreasing r. It follows that the perturbation in energy,

|[pic] |(9) |

is more negative for lower angular momenta. This explains why the nS level is lower than the nP level in Alkali metals (n=2 for Lithium and n=3 for Sodium, etc.)..

problem

Briefly explain why the outer electron S wave is lower than the outer electron P wave in Alkali atoms.

Normal Zeeman Effect

When a hydrogen atom is placed in a strong uniform magnetic field B, certain energy levels “split” into multiple levels. The interaction of the magnetic field with the orbiting electron causes perturbations of the original energy levels that depend on the angular momentum. We note that this “normal” Zeeman effect neglects the contribution of electron spin.

You will recall that a current I experiences a force when it is placed in a magnetic field. The magnetic moment for a current loop of area A is defined by [pic]. The classical value for the electron current is

|[pic] |(10) |

where L is the angular momentum and the energy of the interaction with the field is given by [pic]. If the direction of B is chosen to be the z-direction, then

|[pic] |(11) |

problem

Calculate the energy perturbations for the normal Zeeman effect. [pic]

Notice that each [pic] level is now split into 2[pic]+1 levels by the magnetic field. This is a special case of a general phenomenon. Levels that are degenerate in energy are often perturbed differently and they acquire different energies. We say that the degenerate level is split by the interaction.

*It is always true that (AB...)† = ...B†A† and that (A† ) † = A. Then [pic]

-----------------------

[pic]

[pic]

[pic]

“ket”

what we know

“bra”

what we want to know

Vo

A exp(ikx)

B exp(-ikx)

incident

reflected

C exp(iKx)

transmitted

[pic]

H

O

N

C

C

H

H

H

H

C

C

O

N

O

C

O

O

O

O

................
................

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