1. 3D Schrodinger Equation
Simply substitute momentum operator
do particle in box, HO (quickly), and H atom
added dimensions give more quantum numbers. Can have degeneracies (more than 1 state with same energy). Added complexity. “eta” is really “hbar” thoughout
Solve by separating variables

2. If V well-behaved can separate further: V(r) or Vx(x)+Vy(y)+Vz(z)
If V well-behaved can separate further: V(r) or Vx(x)+Vy(y)+Vz(z). Looking at second one:
LHS depends on x,y RHS depends on z
S = separation constant. Repeat for x and y

3. Example: 2D (~same as 3D) particle in a Square Box
solve 2 differential equations and get
symmetry as square. “broken” if rectangle

4. 2D gives 2 quantum numbers.
Level nx ny Energy E0 E0 E0 E0

5. for degenerate levels, wave functions can mix (unless “something” breaks degeneracy: external or internal B/E field, deformation….)
this still satisfies S.E. with E=5E0. Finite wells can be 2D or 3D but more complicated than 1D. Usually do in R-angle space. HO can also be 2 or 3D and more likely than 3D particle in a box.

6. 3D Harmonic oscillator w = omega and h is hbar

7. Spherical Coordinates
Can solve S.E. if V(r) function only of radial coordinate
volume element is

8. Spherical Coordinates
solve by separation of variables
multiply each side by

9. Spherical Coordinates-Phi
Look at phi equation first. Have separation constant
constant (knowing answer allows form)
must be single valued
the theta equation will add a constraint on the m quantum number – in spherical euqations are coupled through eigenvalues

10. Spherical Coordinates-Theta
Take phi equation, plug into (theta,r) and rearrange. Have second separation constant
knowing answer gives form of constant. Gives theta equation which depends on 2 quantum numbers.

11. Spherical Coordinates-Theta
Associated Legendre equation. Can use either analytical (calculus) or algebraic (group theory) to solve. 383 will skip though slides will be kept
SKIP: Do analytical. Start with Legendre equation

12. Spherical Coordinates-Theta SKIP
Get associated Legendre functions by taking the derivative of the Legendre function. Prove by substitution into Legendre equation
Note that power of P determines how many derivatives one can do.
Solve Legendre equation by series solution

13. Solving Legendre Equation SKIP
Plug series terms into Legendre equation
let k=j+2 in first part and k=j in second (think of it as having two independent sums). Combine all terms with same power
gives recursion relationship
series ends if a value equals 0  L=j=integer
end up with odd/even (Parity) series

14. Solving Legendre Equation SKIP
Can start making Legendre polynomials. Be in ascending power order
can now form associated Legendre polynomials. Can only have l derivatives of each Legendre polynomial. Gives constraint on m (theta solution constrains phi solution)

15. Spherical Harmonics
The product of the theta and phi terms are called Spherical Harmonics. In book
They hold whenever V is function of only r. Saw related to angular momentum

16. Orbital Angular Momentum
Look at the quantum mechanical angular momentum operator (classically this “causes” a rotation about a given axis)
look at 3 components
operators do not necessarily commute z f

17. Side note Polar Coordinates
Write down angular momentum components in polar coordinates
and with some trig manipulations
but same equations will be seen when solving angular part of S.E. and so
and know eigenvalues for L2 and Lz with spherical harmonics being eigenfunctions

18. 3D Schr. Eqn.-Radial Eqn.
For V function of radius only. Look at radial equation. L comes in from theta equation (separation constant)
can be rewritten as (usually much, much better...)
and then have probability

19. 3D Schr. Eqn.-Radial Eqn.
note L(L+1) term. Angular momentum. Acts like repulsive potential and goes to infinity at r=0 (ala classical mechanics)
energy eigenvalues typically depend on 2 quantum numbers (n and L). Only 1/r potentials depend only on n (and true for hydrogen atom only in first order. After adding perturbations due to spin and relativity, depends on n and j=L+s.

20. Particle in spherical box
Good first model for nuclei
plug into radial equation. Can guess solutions
look first at l=0

21. Particle in spherical box
boundary conditions. R=u/r and must be finite at r=0. Gives B=0. For continuity, must have R=u=0 at r=a. gives sin(ka)=0 and
note “plane” wave solution.

22. Particle in spherical box
ForLl>0 solutions are Bessel functions. Can guess shape, like finite 1D well. For every L, “smaller” box, 1, 2, 3…nodes
energy will depend on both quantum numbers
and so 1s 1p 1d 2s 2p 2d 3s 3d …………….and ordering (except higher E for higher n,l) depending on details
gives what nuclei (what Z or N) have filled (sub)shells being different than what atoms have filled electronic shells. In atoms:
in nuclei (with j subshells)

23. H Atom Radial Function For V =a/r get (use reduced mass)
Laguerre equation. Solutions are Laguerre polynomials. Solve using series solution (after pulling out an exponential factor), get recursion relation, get eigenvalues by having the series end……n is any integer > 0 and L<n. Energy doesn’t depend on L quantum number.
Where fine structure constant alpha = 1/137 used. Same as Bohr model energy

24. H Atom Radial Function
Ignore S/L: Energy doesn’t depend on L quantum number but range of L restricted by n quantum number l<n  n=1 only l=0 1S n= l=0, S 2P n= l=0,1, S 3P 3D
eigenfunctions depend on both n,L quantum numbers. First few:

25. H Atom Wave Functions

26. H Atom Degeneracy
As energy only depends on n, more than one state with same energy for n>1 (only first order)
ignore spin for now
Energy n l m D
-13.6 eV (S)
-3.4 eV
1(P) ,0,
1 Ground State
4 First excited states
9 second excited states
-1.5 eV
,0,
2(D) ,-1,0,1,

27. Probability Density P is radial probability density
small r naturally suppressed by phase space (no volume)
can get average, most probable radius, and width (in r) from P(r).

28. Most probable radius For 1S state
Bohr radius (scaled for different levels) is a good approximation of the average or most probable value---depends on n and L but electron probability “spread out” with width about the same size

29. Radial Probability Density

30. Radial Probability Density
note # nodes

31. Angular Probabilities
no phi dependence. If (arbitrarily) have phi be angle around z-axis, this means no x,y dependence to wave function. We’ll see in angular momentum quantization
L=0 states are spherically symmetric. For L>0, individual states are “squished” but in arbitrary direction (unless broken by an external field)
Add up probabilities for all m subshells for a given L get a spherically symmetric probability distribution

32. Orthogonality each individual eigenfunction is also orthogonal.
Many relationships between spherical harmonics. Important in, e.g., matrix element calculations. Or use raising and lowering operators
example

33. Wave functions build up wavefunctions from eigenfunctions. example
what are the expectation values for the energy and the total and z-components of the angular momentum?
have wavefunction in eigenfunction components