PHYS 424 Summary

Probability, the Wave Function, and What to Calculate

Classical and Quantum "States"

Classical	Quantum
Can know position at given time	Can know probability of being at a position at a given time
r(t)	P(r,t) = \|y(r, t)\|² with y_tot(r, t) = y₁(r, t) + y₂(r, t) + ...

Normalization
Particle must be somewhere, so ò dV P(r, t) = 1, i. e. ò dV |y(r, t)|² = 1

Experimental information about a free particle

Classical	Quantum
p = constant	y = e^{ik.r - iwt} with p = hbar k and E = hbar w = p² / (2m)

Basic Equation

Classical	Quantum
- ÑV(r) = m d²r/dt²	Hy_n = i hbar ¶y_n/¶t = E_n y_n [static case] H \|n> = E_n \|n> where H = p²/(2m) + V(r)

Calculating experimental quantities

Classical	Quantum
v(t) = ò ^t a(t') dt' r(t) = ò ^t v(t') dt' E = (1/2)mv² = p²/(2m) p = mv	<Q_operator(t)> = ò y_n^* Q y_n dV = <n \| Q \| n >

Relation between Classical and Quantum Mechanics
- Quantum and classical operators obey similar equations, i.e. L = r x p in both systems, but watch out for the order of noncommuting factors in quantum equations
- Generally <Q(t)> obeys classical equations
Expansions in Eigenstates of Various Operators
- If y_n [ |n> ] are a complete set of states, any y [ | y > ] may be expanded as
  | y > = S_n a_n | n > with a_n = < n | y > , i. e. a_n = ò y_n^* y dV
- Note that if the y_n are eigenstates of some operator Q with eigenvalues Q_n, the probability of getting Q_n when measuring Q in |y> is |a_n|².
- Then if <n | H | m > = ò y_n^* H y_m dV
  S_m < n | H | m > <m | y > = E_n<n | y >
  which you can regard as a matrix eigenvalue equation of H a = E a for the E_n and the coefficients < n | y > = a_n, the a_n regarded as a column matrix. Then just to be a bit more abstract we can call the column matrix y_n, so that the matrix equation also reads H y_n = E_n y_n
Commutators
Partial Differential Equations
Formulas for Spherical Coordinates
Specific Systems and Operators
- One-dimensional Systems
- Angular Momentum and Spin
- Three-dimensional Systems
Perturbation Theory