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!

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

Schroedinger Equation

i

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

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

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

ML1: Problem 2.2, p.24

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

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

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

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

2 22 2

2

1

2 2

dE m x

m dx

HO: Tipler’s approach: Verify solution to SE:

HO: Tipler’s approach..

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

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

Harmonic oscillator: Prob.2.13 Worksheet

ML2: HO, Prob. 2.14 Worksheet

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

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

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