Heisenberg Uncertainty Principle (Mathematical Derivation)

In this article, we will discuss Heisenberg Uncertainty Principle (Mathematical Derivation). So, let’s get started.

Heisenberg Uncertainty Principle (Mathematical Derivation)

Here we give a brief derivation and discussion of Heisenberg’s uncertainty principle. The derivation follows closely that given in Von Neumann’s Mathematical Foundations of Quantum Mechanics. The system we deal with is one dimensional with coordinate X ranging −∞, +∞. The uncertainty principle is a direct consequence of the commutation rule

[X, P] = ih . (1)

While we need this operator equation to derive a concrete result, the general idea is present in any system where there are plane waves. A physically realizable wave is always in the form of a wave packet which is finite in extent. A wave packet is built up by superposing waves with definite wave number. By simple Fourier analysis, a highly localized packet will require a wide spread of wave vectors, whereas a packet with a large spacial extent can be composed of wave numbers quite close to a specific value.

We will carry out the derivation, following Von Neumann, using standard Hilbert space notation instead of Dirac notation. For expected values we write

< P >≡ P¯ = (Ψ, PΨ), and < X >≡ X¯ = (Ψ, XΨ).

Recall that the length of a vector in Hilbert space is defined by
||Ψ|| ≡ √(Ψ, Ψ)

We will also need the uncertainty in X and P. For any observable O, the standard definition of uncertainty is
∆O ≡√(Ψ,(O2 − O’2)Ψ),

which is a measure of the “spread” of values of O around its mean or expected value.

To make progress in the derivation, we need an expression which can be manipulated into one involving [X, P]. The expression used by Von Neumann is (XΨ, PΨ). Using the fact that X and P are self-adjoint, can write

2iIm[(XΨ, PΨ)] = (XΨ, PΨ) − (PΨ, XΨ). (2)

Now we using self-adjointness again, we can move all operators to the right slot, giving.

(XΨ, PΨ) − (PΨ, XΨ) = (Ψ, XPΨ) − (Ψ, P XΨ).

Next we use the X, P commutator and we have

(XΨ, PΨ) − (PΨ, XΨ) = (Ψ, [X, P]Ψ) = ih'(Ψ, Ψ) = ih, . (3)

where in the last equality we have assumed a normalizable state with (Ψ, Ψ) = 1. Comparing Eqs.(2) and (3) we now have

Im[(XΨ, PΨ)] = h/2. (4)

So far we have made no approximations. Eq.(4) is exact. To get the uncertainty relation, we now make use of some inequalities. First we make use of the fact that the imaginary part of a complex number is less than the absolute value of the complex number. For us this says

Im[(XΨ, PΨ)] ≤ |(XΨ, PΨ)|,

which implies using Eq.(4) that

|(XΨ, PΨ)| ≥ h/2. (5)

We now have an inequality involving X and P which came directly from the commutator

Eq.(1), but it involves X and P in the same matrix element. To separate them, we nowuse the Schwarz inequality, which states

|(Ψ, Φ)| ≤ ||Ψ|| · ||Φ||,

which is the Hilbert space version of the familiar statement that the dot product of two vectors is ≤ the product of their lengths. For us, the vectors we will apply the Schwarz inequality to are XΨ, and PΨ. We have

|(XΨ, PΨ)| ≤ ||XΨ|| · ||PΨ||,
so we now have

||XΨ|| · ||PΨ|| ≥ h/2. (6)

This is almost what we want, but makes no reference to the expected values X¯ and P, ¯ both of which come into their respective uncertainties. To remedy this we define

X′ = X − X’I, P′ = P − P’I

These operators satisfy the basic commutation rule,

[X′, P′] = ih.

All the steps in our derivation hold if we replace X by X′, and P by P′, This allows us to write

||X′Ψ|| · ||P′Ψ|| ≥ h/2. (7)

This really is the uncertainty principle. To see this, we write
||X′Ψ||² = (X′Ψ, X′Ψ) = (Ψ, X′X′Ψ) = (Ψ,(X² − 2XX’ + X’²)Ψ)

= (Ψ,(X² − X’²)Ψ) = (∆X)²,

and likewise for ||P′Ψ||2. Using these results we finally have

∆X · ∆P ≥h/2

This completes the mathematical derivation of the uncertainty principle