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Schrodinger Wave Equation

- Schrodinger equation is the first (and easiest)
- works for non-relativistic spin-less particles

(spin added ad-hoc) - guess at form conserve energy, well-behaved,

predictive, consistent with lh/p - free particle waves

Schrodinger Wave Equation

- kinetic potential total energy K U E
- with operator form for momentum and K gives

Giving 1D time-dependent SE

For 3D

Operators

- Operators transform one function to another. Some

operators have eigenvalues and eigenfunctions

Only some functions are eigenfunctions. Only

some values are eigenvalues

In x-space or t-space let p or E be represented

by the operator whose eigenvalues are p or E

Interpret wave function as probability amplitude

for being in interval dx

Example

No forces. V0 solve Schr. Eq

Find average values

(No Transcript)

Solving Schrodinger Equation

- If V(x,t)v(x) than can separate variables

G is separation constant valid any x or t

Gives 2 ordinary diff. Eqns.

GE if 2 energy states, interference/oscillation

1D time independent Scrod. Eqn.

Solve know U(x) and boundary conditions want

mathematically well-behaved.

No discontinuities. Usually except if V0 or y

0 in certain regions

Solutions to Schrod Eqn

- Gives energy eigenvalues and eigenfunctions (wave

functions). These are quantum states. - Linear combinationsof eigenfunctions are also

solutions

Solutions to Schrod Eqn

- Linear combinationsof eigenfunctions are also

solutions. Asuume two energies - assume know wave function at t0
- at later times the state can oscillate between

the two states

The normalization of a wave function doesnt

change with time (unless decays). From Griffiths

Use S.E. to substitute for

J(x,t) is the probability current. Tells rate at

which probability is flowing past point

x substitute into integral and evaluate

The wave function must go to 0 at infinity and

so this is equal 0

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