Time-independent Schrodinger eqn QM Ch.2, Physical Systems, 12.Jan.2003 EJZ

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Time-independent Schrodinger eqnQM Ch.2,
Physical Systems, 12.Jan.2003 EJZ
Assume the potential V(x) does not change in
time. Use separation of variables and
boundary conditions to solve for y. Once you know
y, 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
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Stationary States - introduction

• If evolving wavefunction Y(x,t) y(x) f(t)
• can be separated, then the time-dependent term
  satisfies
• (ML1 will show - class solve for f)
• Separable solutions are stationary states...

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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) yi eigenfunctions corresponding to each
  allowed energy eigenvalue Ei.
• General solution to SE is 2.14

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ML1 Stationary states are separable

• Guess that SE has separable solutions Y(x,t)
  y(x) f(t)
• sub into SESchrodinger Eqn
• Divide by yf
• LHS(t) RHS(x) constantE. Now solve each
  side
• You already found solution to LHS f(t)_________
• RHS solution depends on the form of the potential
  V(x).

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ML1 Problem 2.2, p.24
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Now solve for y(x) for various V(x)

• Strategy
• draw a diagram
• write down boundary conditions (BC)
• think about what form of y(x) will fit the
  potential
• find the wavenumbers kn2 p/l
• find the allowed energies En
• sub k into y(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(0ltxlta) 0, V? outside

• What is probability of finding particle outside?
• Inside SE becomes
• Solve this simple diffeq, using Ep2/2m,
• y(x) A sin kx B cos kx apply BC to find A
  and B
• Draw wavefunctions, find wavenumbers kn a ?np
• find the allowed energies
• sub k into y(x) and normalize
• Finally, the wavefunction is

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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
  x0.
• Preview discuss similarities and differences

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Ex Harmonic oscillator V(x) 1/2 kx2

• Tiplers approach Verify that y0A0e-ax2 is a
  solution
• Analytic approach (2.3.2) rewrite SE diffeq and
  solve
• Algebraic method (2.3.1) ladder operators

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HO Tiplers approach Verify solution to SE

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HO Tiplers approach..
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HO analytically solve the diffeq directly

• Rewrite SE using
• At large xx, has
  solutions
• Guess series solution h(x)
• Consider normalization and BC to find that
  hnan Hn(x) where Hn(x) are Hermite polynomials
• The ground state solution y0 is the same as
  Tiplers
• Higher states can be constructed with ladder
  operators

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HO algebraically use a to get yn

• Ladder operators a generate higher-energy
  wave-functions from the ground state y0.
• Work through Section 2.3.1 together
• Result
• Practice on Problem 2.13

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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 V0

• Looks easy, but we need Fourier series
• If it has a definite energy, it isnt
  normalizable!
• No stationary states for free particles
• Wave functions vg 2 vp, consistent with
  classical particle check this.

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Finite square well V0 outside, -V0 inside

• BC y 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 y(x)
• k, y(x), and E depend on V(x) (shape BC)
• wavefunctions oscillate as eiwt
• wavefunctions can spill out of potential wells
  and tunnel through barriers