Schrödinger Wave Equation

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Reading:Course packet Ch 5.7

SCHROEDINGER WAVE EQUATION

(both images from Wikipedia.com)

Schrödinger Wave Equation

2Schrödinger Wave Equation:

More-or-less "god-given", just like Newton's law F(x) = dp/dt.It works, and if we can show that it fails, we'll refine or discard.

You learned in Spins that the solution to the time-dependent SE is, in terms of the eigenstates of the Hamiltonian:

where

where

Notice the parallels to the rope problem we solved last week?

Schrödinger Wave Equation

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How did this solution to the SWE come about?

How do we solve this for (x,t)? The operator H, the Hamiltonian or energy operator, is clearly special. So representing the kets or the wave function in the energy basis for expanding a general wave function is important! Note that the energy eigenstates are time INdependent if H is time independent. Write down the eigenvalue equation for the the Hamiltonian:

We’ve already solved this equation for the energy eigenstates and energy eigenvalues for the case of the infinite and finite quare wells, so think of those values of E and those functions (x) to be concrete.

Schrödinger Wave Equation

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What follows is cast in terms of wave functions to give you practice, but in fact the ket notation can be used just as easily. The ket version of this discussion is in your text (Ch. 3) and was discussed in Spins.

Expand the wave function in the energy basis. The time dependence comes from coefficients only:

Substitute into

Eigenfunctions of H

Schrödinger Wave Equation

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What follows is cast in terms of wave functions to give you practice, but in fact the ket notation can be used just as easily.

Schrödinger Wave Equation

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Keep working …

So finally …

Schrödinger Wave Equation

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So we have solved Schrödinger's time dependent equation! We know the time dependence, because we already know the eigenstates and energies, which we obtained from solving the eigenvalue equation.

Schrödinger Wave Equation

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There are usually many functions that solve the eigenvalue problem, each function with its own associated eigenvalue. Label them with index n.

In general, the state of particle is NOT an eigenstate of the Hamiltonian, but a superposition of eigenstates …

Very important that the energy in the exponent matches the corresponding eigenstate.

Time dependence

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n =1,2,3,4,5… for x<-a and x>a

Time dependence

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n =1,2,3,4,5… for x<-a and x>a

Time dependence

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At t = 0:

Time dependent state:

In abstract bra-ket notation:

Notice that the constant bn in the previous slide has changed to cn. I used bn before instead of cn to avoid confusion with a time dependent cn(t), but it's so conventional to use cn, that I slipped back to it!

Time dependence

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In position wave function representation:At t = 0:

Time dependent state:

For specific case of the particle in the infinite square well:

Time dependence

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Class activity:• Write short Maple program to animate a general state probability density. How do probability densities and expectation values (qualitative) change with time?• First pick a single eigenstate (any one will do)• Pick a superposition that is a sum of an even and and odd state. • Pick a superposition that is a sum of two even or two odd states.

Time dependence

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Discussion:• Probability density always positive?• What is a stationary state?• Discuss time evolution. • Discuss expectation value of x. • Discuss simple model of atomic transitions

Schrödinger Wave Equation

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Example: V(x) = 0 (free particle)

1st term is traveling wave (to right) and 2nd is traveling to left. Same eigenvalue E - “degenerate”

Schrödinger Wave Equation

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Note the relationship between and k for a free particleDISPERSION RELATION:

Phase velocity Group velocity

Not the same!!!

Schrödinger Wave Equation

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• Time dependent Schrödinger equation• Energy eigenvalue equation (time independent SE)• Eigenstates • Time dependence• (Connection to separation of variables)• Mathematical representations of the above

SCHROEDINGER WAVE EQUATION REVIEW

Schrödinger Wave Equation

18The following treatment parallels the discussion of the 1-D wave equation more closely, but I like the approach earlier better.Schrödinger wave equation:

As in 1-D non-dispersive eqn, use technique of separation of variables.

is function of x only

Schrödinger Wave Equation

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What shall we call the separation constant?

Schrödinger Wave Equation

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Eigenvalue problem! Solutions (and there are many in general) depend on particular V(x). E represents the energy of a particle in that particular eigenstate, and there are many E values.

Most general solution is: