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Time-independent Schrodinger eqnQM Ch.2, Physical Systems, 12.Jan.2003 EJZ
2 2
2
2 2
2
( , ) ( , )( , ) ( , )
2
( )( ) ( ) ( )
2
x t x ti V x t x t
t m x
d xE x V x x
m dx
Assume the potential V(x) does not change in time. Use
* separation of variables and
* boundary conditions to solve for .
Once you know , you can find any expectation value!
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Outline:• “Derive” Schroedinger Eqn (SE)
• Stationary states
• ML1 by Don and Jason R, Problem #2.2
• Infinite square well
• Harmonic oscillator, Problem #2.13
• ML2 by Jason Wall and Andy, Problem #2.14
• Free particle and finite square well
• Summary
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Schroedinger Equation
i
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Stationary States - introduction
If evolving wavefunction (x,t) = (x) f(t)
can be separated, then the time-dependent term satisfies
(ML1 will show - class solve for f)
Separable solutions are stationary states...
i
1 dfi Ef dt
2 2
2
( , ) ( , )( , ) ( , )
2
x t x ti V x t x t
t m x
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Separable solutions:
(1) are stationary states, because
* probability density is independent of time [2.7]
* therefore, expectation values do not change
(2) have definite total energy, since the Hamiltonian is sharply localized: [2.13]
(3) i = eigenfunctions corresponding to each allowed energy eigenvalue Ei.
General solution to SE is [2.14]
2 2( , ) ( )x t x
2 0H
1
( , )ni E t
n nn
x t c e
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ML1: Stationary states are separableGuess that SE has separable solutions (x,t) = (x) f(t)
sub into SE=Schrodinger Eqn
Divide by f:
LHS(t) = RHS(x) = constant=E. Now solve each side:
You already found solution to LHS: f(t)=_________
RHS solution depends on the form of the potential V(x).
t
2
2x
2
22
i Vt x
2 2
22
dV E
m dx
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ML1: Problem 2.2, p.24
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Now solve for (x) for various V(x)
Strategy:
* draw a diagram
* write down boundary conditions (BC)
* think about what form of (x) will fit the potential
* find the wavenumbers kn=2
* find the allowed energies En
* sub k into (x) and normalize to find the amplitude A
* Now you know everything about a QM system in this potential, and you can calculate for any expectation value
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Square well: V(0<x<a) = 0, V= outside
What is probability of finding particle outside?
Inside: SE becomes
* Solve this simple diffeq, using E=p2/2m,
* (x) =A sin kx + B cos kx: apply BC to find A and B
* Draw wavefunctions, find wavenumbers: kn a= n
* find the allowed energies:
* sub k into (x) and normalize:
* Finally, the wavefunction is
2 2
22
dE
m dx
p k
22
2
( ) 2,
2n
nE A
ma a
2( ) sinn
nx x
a a
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Square well: homework
2.4: Repeat the process above, but center the infinite square well of width a about the point x=0.
Preview: discuss similarities and differences
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Ex: Harmonic oscillator: V(x) =1/2 kx2
• Tipler’s approach: Verify that 0=A0e-ax^2 is a solution
• Analytic approach (2.3.2): rewrite SE diffeq and solve• Algebraic method (2.3.1): ladder operators
2 22 2
2
1
2 2
dE m x
m dx
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2 22 2
2
1
2 2
dE m x
m dx
HO: Tipler’s approach: Verify solution to SE:
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HO: Tipler’s approach..
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HO analytically: solve the diffeq directly
Rewrite SE using
* At large ~x, has solutions
* Guess series solution h()
* Consider normalization and BC to find that hn=an Hn() where Hn() are Hermite polynomials
* The ground state solution 0 is the same as Tipler’s
* Higher states can be constructed with ladder operators
2
22
2, ,
m d Ex K K
d
2-a / 20 0( )=A e
22
2
d
d
2- / 2( )=h( )e
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HO algebraically: use a± to get n
Ladder operators a± generate higher-energy wave-functions from the ground state 0.
Work through Section 2.3.1 together
Result:
Practice on Problem 2.13
2
122
1
2
( ) ,m
xnn n n
da im x
i dxm
A a e with E n
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Harmonic oscillator: Prob.2.13 Worksheet
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ML2: HO, Prob. 2.14 Worksheet
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Ex: Free particle: V=0
• Looks easy, but we need Fourier series
• If it has a definite energy, it isn’t normalizable!
• No stationary states for free particles
• Wave function’s vg = 2 vp, consistent with classical particle: check this.
2
2
k
m
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Finite square well: V=0 outside, -V0 inside
• BC: NOT zero at edges, so wavefunction can spill out of potential
• Wide deep well has many, but finite, states• Shallow, narrow well has at least one bound state
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Summary:
• Time-independent Schrodinger equation has stationary states (x)
• k, (x), and E depend on V(x) (shape & BC)
• wavefunctions oscillate as eit
• wavefunctions can spill out of potential wells and tunnel through barriers