FOUNDATIONS OF QUANTUM MECHANICS

98 CHAPTER IV. THE COPENHAGEN INTERPRETATION
IV. 5. 2
THE STANDARD UNCERTAINTY RELATIONS
If ψ ∈ L 2 (R) is the normalized wave function of a physical system in the q - language,
with ∥ψ∥ = 1, the wave function ˜ψ(p) in the p - language is its Fourier transform
˜ψ(p) =
∫
1
√
2 π
R
e − i p q
ψ(q) dq, (IV. 27)
and its inverse Fourier transform is
∫
1
ψ(q) = √ e i p q
˜ψ(p) dp. (IV. 28)
2 π
R
The norm is invariant under Fourier transformations, therefore ∥ ˜ψ∥ = 1.
The standard deviation of position in a state |ψ⟩, ∆ ψ Q, is defined as
∫
( ∫ ) 2.
(∆ ψ Q) 2 = ⟨Q 2 ⟩ ψ − ⟨Q⟩ ψ 2 = q 2 |ψ(q)| 2 dq − q |ψ(q)| 2 dq (IV. 29)
R
R
Likewise, for momentum, ∆ ψ P , we have
(∆ ψ P ) 2 = ⟨P 2 ⟩ ψ − ⟨P ⟩ ψ
2
∫
= − 2 ψ ∗ (q) d2 ψ(q)
( ∫
R dq 2 dq − − i ψ ∗ (q) dψ(q) ) 2
dq
R dq
∫
= p 2 | ˜ψ(p)|
( ∫ 2. 2 dp − p | ˜ψ(p)| dp) 2 (IV. 30)
R
R
Without loss of generality we can assume ⟨P ⟩ and ⟨Q⟩ to equal 0, so that
of 1 2
(∆ ψ P ) 2 = − 2 ∫
R
ψ ∗ (q) d2 ψ(q)
dq 2 dq =
∫
R
p 2 | ˜ψ(p)| 2 dp. (IV. 31)
If the wave function ψ (q) is a Gaussian wave packet, the product takes on the minimum value
. An example is the ground state of the one - dimensional harmonic oscillator having mass m,
ϕ 0 (q) =
( m ω0
π
) 1
4 e − m ω q2
2 , (IV. 32)
with energy E 0 = 1 2 ω 0.
Before interpreting the Kennard inequality (IV. 26), we give a still more general inequality, derived
by Schrödinger (1930). Consider two arbitrary self - adjoint operators A and B acting on a Hilbert
space H. Define, for a pure state |ψ⟩ ∈ H, the following operators:
A ψ := A − ⟨A⟩ ψ 11 and B ψ := B − ⟨B⟩ ψ 11. (IV. 33)
The expectation values of these operators are, in the state |ψ⟩, equal to 0,
⟨A ψ ⟩ ψ = ⟨B ψ ⟩ ψ = 0. (IV. 34)