The_Cambridge_Handbook_of_Physics_Formulas

4.2 Quantum definitions
91
Operators
Hermitian
conjugate
operator
Position
operator
Momentum
operator
Kinetic energy
operator
Hamiltonian
operator
Angular
momentum
operators
∫
∫
(âφ) ∗ ψ dx =
φ ∗ âψ dx (4.17)
ˆ x n = x n (4.18)
pˆ
n x = ¯hn ∂ n
i n ∂x n (4.19)
∂ 2
ˆT = − ¯h2
2m ∂x 2 (4.20)
∂ 2
â Hermitian conjugate
operator
ψ,φ normalisable functions
∗
x,y
complex conjugate
position coordinates
n arbitrary integer ≥ 1
p x
T
¯h
m
momentum coordinate
kinetic energy
(Planck constant)/(2π)
particle mass
Ĥ = − ¯h2
2m ∂x 2 +V (x) (4.21) H Hamiltonian
V potential energy
Lˆ
z = ˆx pˆ
y −ŷpˆ
x (4.22)
Lˆ
2 2 2 2
= Lˆ
x + Lˆ
y + Lˆ
z (4.23)
L z
L
angular momentum along
z axis (sim. x and y)
total angular momentum
4
Parity operator ˆPψ(r)=ψ(−r) (4.24)
ˆP
r
parity operator
position vector
Expectation value
∫
Expectation 〈a〉 = 〈â〉 = Ψ ∗ âΨdx (4.25)
value a = 〈Ψ|â|Ψ〉 (4.26)
Time
dependence
Relation to
eigenfunctions
Ehrenfest’s
theorem
d
dt 〈â〉 = ī 〈 〉 ∂â
h 〈[Ĥ,â]〉+ ∂t
if âψ n = a n ψ n and Ψ = ∑ c n ψ n
(4.27)
then 〈a〉 = ∑ |c n | 2 a n (4.28)
〈a〉
â
Ψ
x
t
¯h
ψ n
a n
n
c n
expectation value of a
operator for a
(spatial) wavefunction
(spatial) coordinate
time
(Planck constant)/(2π)
eigenfunctions of â
eigenvalues
dummy index
probability amplitudes
m d m particle mass
〈r〉 = 〈p〉 (4.29)
dt r position vector
d
dt 〈p〉 = −〈∇V 〉 (4.30) p momentum
V potential energy
a Equation (4.26) uses the Dirac “bra-ket” notation for integrals involving operators. The presence of vertical bars
distinguishes this use of angled brackets from that on the left-hand side of the equations. Note that 〈a〉 and 〈â〉 are
taken as equivalent.