A
Non-relativistic Quantum Mechanics
This appendix is intended as a very terse ‘revision’ summary of those aspects
of non-relativistic quantum mechanics that are particularly relevant for this
book. A fuller account may be found in Mandl (1992), for example.
Natural units ħ = c = 1 (see appendix B).

Fundamental postulate of quantum mechanics:

[ˆ p i , x ˆ j ] = −iδ ij .
(A.1)
Coordinate representation:
p ˆ = −i∇
(A.2)
∂ψ(x, t)
ˆ
Hψ(x, t) = i
.
(A.3)
∂t
Schr¨ odinger equation for a spinless particle:
2
p ˆ
H ˆ =
+ V ˆ
(A.4)
2m
and so
(
)
1
∂ψ(x, t)
−
∇
2 + V ˆ (x, t) ψ(x, t) = i
.
(A.5)
2m
∂t
Probability density and current (see problem 3.1 (a)):
ρ = ψ
∗ ψ = |ψ|
2
≥ 0
( A . 6 )
j =
1 [ψ
∗ (∇ψ) − (∇ψ
∗ )ψ]
( A . 7 )
2mi
with
∂ρ + ∇ · j = 0.
(A.8)
∂t
Free-particle solutions:
φ(x, t) = u(x)e
−iEt
(A.9)
ˆ
H 0 u = Eu
(A.10)
where
H ˆ 0 = H ˆ (V ˆ = 0).
(A.11)
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