Download Physics 341: Useful Quantum Facts Particle in a Box Hamiltonian: H =

Physics 341: Useful Quantum Facts
Particle in a Box
Hamiltonian: H =
Solutions: ψn (x) =
Energy: En =
2
h̄2 kn
2m
p2
2m
q
=
2
2
h̄ d
= − 2m
dx2 ;
2
L
sin kn x;
Boundary conditions: ψ(0) = 0, ψ(L) = 0
kn =
nπ
L ;
h̄2 π 2 2
2mL2 n
3
d3 n = Vhd3 p (3-dimensions)
Density of states: Momentum form: dn = Ldp
h (1-dimension);
1
p
3
2
Energy form in 3-dimensions: Use dE = m
dp to find dn = 4πV
h3 (2m E) dE
References: 1. Thornton and Rex, Section 6.3. For density of states, see discussion leading to Eq. 9.43 which
includes a extra factor of 2 for the electron’s spin multiplicity.
2. N. Zettili, Section 4.6; density of states just given in Eq. 10.134.
Harmonic Oscillator
p2
1
2
2m + 2 kx
2
2
h̄ d
1
2
= − 2m
; Boundary/symmetry conditions: ψ(−x) = ±ψ(x), ψ(∞) = 0
dx2 + 2 kx
q
√
2
; ζ = x α, and Hn (ζ) is the Hermite polynomial of order n.
Solutions: ψn (ζ) = Hn (ζ)e−ζ /2 ; where α = mk
h̄2
p
Energy: En = h̄ω(n + 21 ); ω = k/m
References:1. Thornton and Rex, Section 6.6. See also Section 10.1, page 337, for application to molecular
vibrations.
2. N. Zettili, Section 4.8
Hamiltonian: H =
Orbital Angular Momentum and Rigid Rotator
2
2
∂
∂
∂2
Hamiltonian: H = L2I = h̄2I sin1 θ ∂θ
sin θ ∂θ
+ sin12 θ ∂φ
;
2
m2
where I = µR2 , µ = mm11+m
, and R is the separation of the masses.
2
Solution: Ylm (θ, φ) (spherical harmonics); l = 0, 1, . . .; −l ≤ m ≤ +l,
multiplicity (number of m-states): 2l + 1
2
Energy: El = h̄ l(l+1)
2I
References: 1. See Thornton and Rex, Section 7.2, page 244, for development of spherical harmonics (Table
7.2). For application to molecular rotations see Section 10.1, page 336, Eq. 10.2.
2. N. Zettili, Section 5.7.2-3; problem 5.3, page 311.
Angular Momentum
Addition of angular momenta: J~ = J~1 + J~2 ;
Eigenstates: J 2 |jmj >= h̄2 j(j + 1)|jmj >; Jz |jmj >= mj |jmj >;
Example: orbital angular momentum |j = l, mj = ml >→ Ylm (θ, φ)
Quantum numbers arising from addition of angular momentum: |j1 − j2 | ≤ j ≤ j1 + j2 ;
P
multiplicity: j (2j + 1) = (2j1 + 1)(2j2 + 1)
Example: Spin-orbit addition: j1 = l, j2 = s = 21 ; → l − 12 ≤ j ≤ l + 21 → j = l + 21 , l − 12
P
multiplicity: j (2j + 1) = (2(l + 12 ) + 1) + (2(l − 12 ) + 1) = 2(2l + 1) = (2s + 1)(2l + 1)
References: 1. Thornton and Rex, Section 8.2.
2. N. Zettili, Section 7.3.
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