1. Solving Schrodinger Equation: ”Easy” Bound States
If V(x,t)=v(x) than can separate variables
G is separation constant valid any x or t.
note h is really hbar
Gives 2 ordinary diff. Eqns. 1

2. 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 2

3. 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 3

4. 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. 4

5. 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. 5

6. Boundary condition is that y is continuous:give:
Square Well Potential
Start with the simplest potential
Boundary condition is that y is continuous:give: V
Draw wave functions; readily get wavelength
p=h/l
-a/ a/2 6

7. Infinite Square Well Potential
Solve S.E. where V=0
Boundary condition quanitizes k/E, 2 classes
Even
y=Bcos(knx)
kn=np/a
n=1,3,5...
y(x)=y(-x)
Odd
y=Asin(knx)
kn=np/a
n=2,4,6...
y(x)=-y(-x)
Parity operator
Note:shift x-axis 7

8. 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 8

9. 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 eV about 0.01 eV? 9

10. Example 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 (essentially dot product) 10

11. Example Particle in box with width a and a wavefunction of
psi(x) = sqrt(4/5)u_1(x) + sqrt(1/5)u_2(x)
What is <E>?
<E> = 4/5 E_ /5 E_2 = h^2/8ma^2(4/5 + 1/5*4)
Draw the wavefunction. Is <x> non-zero? 11

12. Finite Square Well Potential
For V=finite “outside” the well. Solutions to S.E. inside the well the same. Have different outside. The boundary conditions (wavefunction and its derivative continuous) give quantization for E<V 0
longer wavelength, lower Energy. Finite number of energy levels
Outside:

13. Boundary Condition
Want wavefunction and its derivative to be continuous
often a symmetry such that solution at +a also gives one at -a
Often can do the ratio (see book which solves) and that can simplify the algebra

14. Finite Square Well Potential:mostly skip
Equate wave function at boundaries
And derivative

15. Finite Square Well:mostly skip
Harris does algebra. 2 classes. Solve numerically
k1 and k2 both depend on E. Quantization sets allowed energy levels

16. Finite Square Well Potential:remember this
Number of bound states is finite. Calculate
assuming “infinite” well energies. Get n. Add 1
Electron V=100 eV width=0.2 nm
Deuteron p-n bound state. Binding energy 2.2 MeV
radius = 2.1 F (really need 3D S.E………)

17. Finite Square Well Potential:remember this
Can do an approximation by guessing at the
penetration distance into the “forbidden” region. Use to estimate wavelength
Electron V=100 eV width=0.2 nm d

18. Delta Function Potential:skip
d-function can be used to describe potential.
Assume attractive potential V and E<V bound state. Potential has strength l/a. Rewrite Schro.Eq V

19. Delta Function Potential II:skip
Except for x=0 have exponential solutions.
Continuity condition at x=0 is (in some sense) on the derivative. See by integrating S.E in small region about x=0 V

20. Harmonic Oscillators V
F=-kx or V=cx2. Arises often as first approximation for the minimum of a potential well
Solve directly through “calculus” (analytical). Do in 460
Solve using group-theory like methods from relationship between x and p (algebraic). Do in 460 V
Classically F=ma and
Sch.Eq.

21. Harmonic Oscillators-Guess
Can use our solution to finite well to guess at a solution
Know lowest energy is 1-node, second is 2-node, etc. Know will be Parity eigenstates (odd, even functions)
Could try to match at boundary but turns out in this case can solve the diff.eq. for all x - as no abrupt changes in V(x) V y

22. Solve Hermite Equation
Get recursion relationship. Skip
if any al=0 then the series can end (higher terms are 0). Gives eigenvalues
easiest if define odd or even types. Need to know energy eigenvalues, but not how to derive

23. Hermite Equation - wavefunctions
Use recursion relationship to form eigenfunctions. Compare to first 2 for infinite well
First
Second
Third