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3D Schrodinger Equation
Simply substitute momentum operator do particle in box and H atom added dimensions give more quantum numbers. Can have degeneracies (more than 1 state with same energy). Added complexity. Solve by separating variables P D S.E.

LHS depends on x,y RHS depends on 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 P D S.E.

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

2D gives 2 quantum numbers. Level nx ny Energy 1-1 1 1 2E0 1-2 1 2 5E0
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 P D S.E.

Spherical Coordinates
Can solve S.E. if V(r) function only of radial coordinate volume element is solve by separation of variables multiply each side by P D S.E.

Spherical Coordinates-Phi
Look at phi equation first constant (knowing answer allows form) must be single valued the theta equation will add a constraint on the m quantum number P D S.E.

Spherical Coordinates-Theta
Take phi equation, plug into (theta,r) and rearrange knowing answer gives form of constant. Gives theta equation which depends on 2 quantum numbers. Associated Legendre equation. Can use either analytical (calculus) or algebraic (group theory) to solve. Do analytical. Start with Legendre equation P D S.E.

Spherical Coordinates-Theta
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 P D S.E.

Solving Legendre Equation
Plug series terms into Legendre equation let k-1=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 P D S.E.

Solving Legendre Equation
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) P D S.E.

Spherical Harmonics The product of the theta and phi terms are called Spherical Harmonics. Also occur in E&M. They hold whenever V is function of only r. Seen related to angular momentum P D S.E.

3D Schr. Eqn.-Radial Eqn. For V function of radius only. Look at radial equation can be rewritten as (usually much better...) 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). P D S.E.

Particle in spherical box
Good first model for nuclei plug into radial equation. Can guess solutions look first at l=0 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. Supplement 8-B discusses scattering, phase shifts. General terms are P D S.E.

Particle in spherical box
ForLl>0 solutions are Bessel functions. Often arises in scattering off spherically symmetric potentials (like nuclei…..). Can guess shape (also can guess finite well) 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) P D S.E.

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 eigenfunctions depend on both n,L quantum numbers. First few: P D S.E.

H Atom Wave Functions P D S.E.

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

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). (Supplement 8-A) P D S.E.

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 P D S.E.

Radial Probability Density
P D S.E.

Radial Probability Density
note # nodes P D S.E.

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 P D S.E.

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 P D S.E.

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 P D S.E.

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