Schrodinger Wave Equation

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# Schrodinger Wave Equation - PowerPoint PPT Presentation

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 l =h/p free particle waves. Schrodinger Wave Equation.

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Presentation Transcript
Schrodinger Wave Equation
• Schrodinger equation is the first (and easiest)
• guess at form: conserve energy, well-behaved, predictive, consistent with l=h/p
• free particle waves

P460 - Sch. wave eqn.

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:

P460 - Sch. wave eqn.

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

P460 - Sch. wave eqn.

P460 - Sch. wave eqn.

Example

No forces. V=0 solve Schr. Eq

Find average values

P460 - Sch. wave eqn.

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.

P460 - Sch. wave eqn.

G=E 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 V=0 or y =0

in certain regions

P460 - Sch. wave eqn.

Solutions to Schrod Eqn
• Gives energy eigenvalues and eigenfunctions (wave functions). These are quantum states.
• Linear combinationsof eigenfunctions are also solutions

P460 - Sch. wave eqn.

Solutions to Schrod Eqn
• Linear combinationsof eigenfunctions are also solutions. Asuume two energies
• assume know wave function at t=0
• at later times the state can oscillate between the two states

P460 - Sch. wave eqn.

The normalization of a wave function doesn’t 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

P460 - Sch. wave eqn.