7.4The Heisenberg Uncertainty Principle

The more precisely the position is determined, the less precisely the momentum is known in this instant, and vice versa. Werner Heisenberg, 1927.

In §7.3 we proved the Heisenberg Uncertainty Principle for the special case of the harmonic oscillator. Here we show this is a general feature of quantum mechanics. First we recall some basic facts about complex vector spaces.

1.If Ψ and Φ are any two states in our Hilbert space of states H, we have an inner product defined (Ψ, Φ) that satisfies the properties

(a)(Ψ, Φ) = (Φ, Ψ) where z denotes the complex conjugate of z.

(b)(c1Ψ1 + c2Ψ2, Φ) = c1 (Ψ1, Φ) + c2 (Ψ2, Φ) for all states Ψ1, Ψ2 and all complex numbers c1, c2.

(c)The length or norm of the state Ψ is defined to be kΨk2 = (Ψ, Ψ) ≥ 0 with kΨk = 0 if and only if Ψ = 0, the zero vector in H.

2.An operator A is called Hermitian (or self-adjoint) if

(AΨ, Φ) = (Ψ, AΦ)

for all states Ψ, Φ. In quantum mechanics observables are assumed to be Hermitian. Note this makes the expected value of the observable A in state Ψ a real number

hAi := (AΨ, Ψ) = (Ψ, AΨ)

= (AΨ, Ψ)

= hAi.

Sometimes one writes hAiΨ to denote the state in which the expected value is computed.

We now assume we have observables A and B that satisfy the commutation relation

where id is the identity operator and i is the imaginary number, i2 = −1. We showed earlier that in units where ~ = 1 the position and momentum operators satisfy such a commutation relation. For a given state Ψ and observable A we define21

r

q

DE

A = (A − hAi)2 = hA2i − hAi2 ≥ 0.

21In classical probability A is called the standard deviation of A. The quantity A is a measure of the deviation of A from its expected value hAi.

Proof: Let Ψ denote any normalized state vector, i.e. kΨk = 1. Apply (7.29) to Ψ to obtain

hAi2 + t2hBi2 ≤ kAΨk2 − t + t2kBΨk2.

Using the fact that kAΨk2 = (AΨ, AΨ) = (A2Ψ, Ψ) = hA2i (and similarly for B) and the definition of A (and similarly for B), the above inequality can be rewritten as

t2 (ΔB)2 − t + (ΔA)2 ≥ 0.

This holds for all real t. The above is a quadratic polynomial in t that is always nonnegative. This means that the discriminant of the quadratic polynomial must be nonpositive, i.e.

b2 − 4ac ≤ 0. That is,

1 − 4 (ΔA)2 (ΔB)2 ≤ 0

which implies that

A · B ≥ 12

which is what we want to prove.

When A is the position operator and B is the momentum operator we get the Heisenberg

Uncertainty Principle which states

x · p ≥ ~2

where we have returned to physical units. The appearance of Planck’s constant ~ in the right hand side shows that ~ sets the scale for quantum mechanical phenomena.

22Note (ABΨ, Ψ) = (BΨ, AΨ) and (BAΨ, Ψ) = (AΨ, BΨ) since A and B are Hermitian. Also note on the right hand side we used the fact that (Ψ, Ψ) = kΨk2 = 1.