Solving Schrodinger Equation

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# Solving Schrodinger Equation - PowerPoint PPT Presentation

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. 1. Solutions to Schrod Eqn. Gives energy eigenvalues and eigenfunctions (wave functions). These are quantum states.

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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.

1

Solutions to Schrod Eqn

Gives energy eigenvalues and eigenfunctions (wave functions). These are quantum states.

Linear combinations of eigenfunctions are also solutions. For discrete solutions

If H Hermitian

P460 - Sch. wave eqn.

2

G=E if 2 energy states, interference/oscillation

1D time

independent

Scrod. Eqn.

Solve: know U(x) and boundary conditions

want mathematically well-behaved. Do not want:

No discontinuities. Usually

except if V=0 or y =0

in certain regions

P460 - Sch. wave eqn.

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Linear Operators

Operator converts one function into another

an operator is linear if (to see, substitute in a function)

linear suppositions of eigenfunctions also solution if operator is linear……use “Linear algebra” concepts. Often use linear algebra to solve non-linear functions….

P460 - Sch. wave eqn.

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Solutions to Schrod Eqn

Depending on conditions, can have either discrete or continuous solutions or a combination

where Cn and C(E) are determined by taking the dot product of an arbitrary function y with the eigenfunctions u. Any function in the space can be made from linear combinations

P460 - Sch. wave eqn.

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Solutions to Schrod Eqn

Linear combinations of eigenfunctions are also solutions. Assume two energies

assume know wave function at t=0

at later times the state can oscillate between the two states - probability to be at any x has a time dependence

P460 - Sch. wave eqn.

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Example 3-1

Boundary conditions (including the functions being mathematically well behaved) can cause only certain, discrete eigenfunctions

solve eigenvalue equation

impose the periodic condition to find the allowed eigenvalues

P460 - Sch. wave eqn.

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Square Well Potential

Boundary condition is that y is continuous:give:

V

0

-a/2 a/2

P460 - Sch. wave eqn.

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Infinite Square Well Potential

Solve S.E. where V=0

Boundary condition quanitizes k/E, 2 classes

Odd

y=Asin(knx)

kn=np/a

n=2,4,6...

y(x)=-y(-x)

Even

y=Bcos(knx)

kn=np/a

n=1,3,5...

y(x)=y(-x)

P460 - Sch. wave eqn.

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Parity

Parity operator P x  -x (mirror)

determine eigenvalues

even and odd functions are eigenfunctions of P

any function can be split into even and odd

P460 - Sch. wave eqn.

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Parity

If V(x) is an even function then H is also even then H and P commute

and parity is a constant. If the initial state is even it stays even, odd stays odd. Semi-prove:

time development of a wavefunction is given by

do the same for Py when [H,P]=0

and so a state of definite parity (+,-) doesn’t change parity over time; parity is conserved (strong and EM forces conserve, weak force does not)

P460 - Sch. wave eqn.

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Infinite Square Well Potential

Need to normalize the wavefunction. Look up in integral tables

What is the minimum energy of an electron confined

to a nucleus? Let a = 10-14m = 10 F

P460 - Sch. wave eqn.

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Infinite Square Well Density of States

The density of states is an important item in determining the probability that an interaction or decay will occur

it is defined as

for the infinite well

For electron with a = 1mm, what is the number of states within 0.0001 eV about 0.01 eV?

P460 - Sch. wave eqn.

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Example 3-5

Particle in box with width a and a wavefunction of

Find the probability that a measurement of the energy gives the eigenvalue En

With only n=odd only from the symmetry

The probability to be in state n is then

P460 - Sch. wave eqn.

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Free particle wavefunction

If V=0 everywhere then solutions are

but the exponentials are also eigenfunctions of the momentum operator

can use to describe left and right traveling waves

book describes different normalization factors

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