Quantum definitions
Energy levels of a harmonic oscillator
(n+1/2)hbar w
The momentum operator
*P*=(P1,P2,P3) Pj phi(*x*)= -ihbar d/dxjphi(*x*) for j=1,2,3
The position operator
*X*=(X1,X2,X3) Xj phi(*x*) = xj phi(*x*) for j=1,2,3
The probability current, *j*
*j*(*r*,t) = ihbar/2m (phi conj(grad(phi)) - conj(phi) grad(phi)
If phi(*r*,t) = sum over n (an phi(n)(*r*) exp(-iEnt/hbar), what is the probability of measuring the energy of the particle to be En?
*|an|^2 * Since 1 = integral from 0 to a of |phi(x,t)|^2 = sum from m,n=1 to infinity of {conj(am)an exp(-I(En-Em)t/hbar) * integral from 0 to a conj(phi(m)(x))phi(n)(x))dx} = sum from n=1 to inf of |an|^2
The alternating symbol e(ijk)
+1 if ijk is an even permutation of 123 -1 if ijk is an odd permutation of 123 0 otherwise
Energy of a hydrogen atom
-hbar^2 b^2 / (8 m (N+1)^2)
The time independent Schrodinger Equation
-hbar^2/2m grad^2(phi) + Vphi = Ephi
Wave equation
1/v^2 d^2phi/dt^2 = grad^2(phi)
Assuming no degeneracies, what is the dimension on the eigenspace on which J^2=j(j+1)hbar
2j+1 Note this is the then the dimension of H
The state of a particle
A given solution to the stationary state SE
Free particle
A particle subjected to no forces
Coulomb's Law
A point charge e2 at the origin induces an electrostatic force on another point charge e1 at position *r* given by *F* = e1 e2 *r* / (4 pi eps0 r^3) where r = |*r*|
A minimum uncertainty state
A state such that disp(P)disp(X)=h/2
What is the collapse of the wave function?
After measuring the wave function, the wave function collapses into an eigenspace
Central force
Any potential which depends only on r
Angular momentum operators
Any set of operators Ji that satisfy [Ji,Jj] = i hbar sum on k e(ijk) Jk
Why must even series terminate?
Because coefficients behave like that of exp(zeta^2), ie they have the same asymptotic expansion
Time-dependent wave function for a particle in a 1D box
Bsin(n pi x/a) exp(-in^2 pi^2 bar t/2ma^2)
Time-independent wave function for a particle in a 1D box
Bsin(n pi x/a) for 0<x<a
Total energy of a particle
E = 1/2 m |d*r*/dt|^2 + V
Expectation of an observable A in a normalisable state phi
E(phi)(A)=<phi|Aphi>/<phi|phi>
Expectation value of a function of position f(*r*)
E(phi)(f(*r*)) is the integral over R^3 of f(*r*)|phi(*r*,t)|^2
Einstein-Planck relation
E=hbarw
De Broglie's Relations
E=hbarw *p*=hbar*k*
Kinetic energy in terms of momentum
E=|*p*|^2/2m
States of a quantum system
Elements of a complex vector space H
Conservative forces
Forces such that *F*=-grad(V) for some potential V=V(*r*)
The Hamiltonian operator for a particle of mass m moving in a potential V
H:= 1/2m (P1^2+P2^2+P3^2) + V(X1,X2,X3)
Ladder operators
J± = J1 ± i J2
What is the principal quantum number
N+1
Complementary observables
Observables that cannot be measured simultaneously to arbitrary accuracy
Observables of a quantum system
Self adjoint linear transformations A of H (ie A=A* where <A*psi|phi>=<psi|Aphi> for all psi, phi in H)
How to get rotationally symmetric solutions
Setting l=0=m
Quanta
Small packets
Photons
Small packets of light/quanta of energy E=hbarw
What is the degeneracy of the eigenstates of a hydrogen atom?
Sum from l=0 to N of 2l+1 = *(N+1)^2*
Normalised wave function
The integral over R^3 of |phi(*r*,t)|^2 is 1
Normalisable wave function
The integral over R^3 of |phi(*r*,t)|^2 is finite
Ground state energy
The lowest possible energy
E is a non-degenerate energy level
The space of solutions to the stationary state SE with energy E has dimension 1
Energy level E is d-fold degenerate
The space of solutions to the stationary state SE with energy E has dimension d>1
Correspondence principle
The tendency of quantum results to approach those of classical theory for large quantum numbers
What conditions should the solutions to the SE satisfy
The wave function should be a continuous, single valued function The wave function should be normalisable grad(phi) should be continuous everywhere, except where there is an infinite discontinuity in the potential V
Coulomb Potential
V(*r*) = V(r) = e1 e2 / (4 pi eps0 r)
Wave function and energies of a particle in a 3D box
Wave function is a product of three 1D wave functions Energy is the sum of three 1D energies
Spherical harmonic of order l, m
Yl,m(theta, phi) = Plm(theta) exp(i m phi) The eigenvector of L^2 in spherical coordinates, with eigenvalue -l(l+1)
Define the commutator of observables A, B
[A,B]=AB-BA
What is [P,X]?
[P,X]=-ihbarI
What is [Pi,Xj]?
[Pi,Xj]=-ihbar delta(i,j) I
Define a±
a±=P±imwX
Heisenberg's Uncertainty Principle
disp(P)disp(X)≥h/2 with equality iff phi=const x exp(-t/2hbar (x-mu)^2)
Continuity equation
dp/dt + div(*j*) = 0
The time dependent Schrodinger Equation
i hbar dphi/dt = -hbar^2/2m grad^2(phi) + Vphi
What is the azimuthal quantum number?
l
What is the eigenvalue of J^2?
l(l+1)hbar^2
What is the magnetic quantum number?
m
What is the eigenvalue of J3?
m hbar
Degeneracy of the original 2D oscillator potential with w1=w2 (ie R=(n+1)hbarw
n+1
Energies of a particle in a 1D box
n^2 pi^2 hbar^2 /2ma^2
Atomic number
number of protons
Probability density function for the position of the particle
p(*r*,t)=|phi(*r*,t)|^2
Stationary state wave function
phi(*r*) exp(-iEt/hbar)
Time-independent ground state wave function for the harmonic oscillator
phi(0)(x)=a0exp(-mwx^2/2hbar)
Time-dependent normalised ground state wave function for the harmonic oscillator
phi(0)(x,t)=(mw/hbarpi)^(1/4) exp(-(mwx^2+ihbarwt)/2hbar)
Give the form of the wave function for the hydrogen atom (The eigenfunction of the Hamiltonian, J^2 and J3)
phi(N,l,m) ~ f(N,l)(r) exp(-k(N)r) Y(l,m)(theta, phi)
Dispersion of A in a state phi
{E(phi)(A^2)-E(phi)(A)^2}^(1/2)
If phi = sum over n an phi(n) exp(-iE(n)t/hbar), what is the probability of measuring the energy to be En?
|an|^2
If H has an ON basis of eigenvectors of A with eigenvectors a(lambda), what is the probability that measurement of A gives a(lambda) in a state phi?
|c(lambda)|^2 where phi = sum over lambda c(lambda)phi(lambda)
A state phi is normalised
||phi||^2 = <phi|phi> = 1
