LECTURE 3 - Hope College
LECTURE 7
2-D Wave Equation
Vibrating membrane instead of vibrating string
[pic]
where u = u(x, y, t)
Separation of variables:
[pic]
Solve 2 spatial ODE’s
Apply boundary conditions
[pic]
Final Solution
[pic]
Normal Modes
[pic]
Note: nodes are lines for 2-D waves
number of nodes = (n - 1) + (m - 1)
“String Theory Meets Practice as Violinmakers Rethink Their Craft”
Newspaper article about people who are looking at vibrations in a violin’s sound box.
Question: If nodes of 1-D waves are points, nodes of 2-D waves are lines, what are nodes of 3-D waves?
Answer:3-D waves have planes/ curved surfaces as nodes
Now starting modern quantum mechanics
▪ Historical background important
o Gives us perspective
o Intuitive understanding of QM
The Schrodinger Wave Equation
“In this communication, I wish to show…that the usual rules of quantization can be replaced by another postulate, in which there occurs no mention of whole numbers. Instead, the introduction of integers arises in the same natural way as, for example, in a vibrating string, for which the number of nodes is integral. The new conception can be generalized, and I believe that it penetrates deeply into the true nature of the quantum rules.
In this an four other papers, published during the first half of 1926, Schrodinger communicated his wave equation and applied it to a number of problems, including the hydrogen atom, the harmonic oscillator, the rigid rotor, the diatomic molecule…”
Schrodinger Equation
Fundamental Equation of Quantum Chemistry
Schrodinger (1926): If particles possess wave-like properties, then there exists a wave equation which governs their behavior.
Start with 1-D classical wave equation
[pic][pic]
Assume that the spatial solution is independent of time, i.e. separation of variables
[pic]
[pic]
then
[pic]
[pic]
Assumptions:
1) Classical wave equation
2) Time independent spatial solution
3) Wave nature of particles (deBroglie relation)
Born Interpretation of the Wave function
Recall Einstein’s explanation of the photoelectric effect:
The number of photons is related to the intensity of light, which is related to the E-M wave amplitude squared.
The wave function, (, represents the “amplitude” of a particle.
Since intensity is (amplitude)2 for waves,
[pic] is the intensity of a particle.
Note: ( = complex conjugate (i ( - i)
(a + ib)(a – ib) = a2 + b2
Born Interpretation (1926):[pic] is the probability of locating the particle between x and x + dx.
Mathematical Properties of (:
1) ( is a single valued (wave function)
2) ( and (’ are continuous (wave function)
3) ( is normalized, i.e., [pic]
probability of finding particle somewhere is one!
-----------------------
complex
real!
complex conjugate
Energy
(Kinetic and Potential)
Hamiltonian
Operator
Wave function
Time-independent Schrodinger Equation
general temporal solution
unspecified spatial solution
(deBroglie relation)
n=3 m=2
n=1 m=2
n=2 m=1
x
n=1 m=1
-
-
-
-
+
+
-
+
+
+
+
and
Zero along boundaries
a
b
(0,0)
x
y
................
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