quantum mechanics - Heisenberg Uncertainty Principle scientific proof

Heisenberg's uncertainty principle states that:

$$\sigma(x)\sigma( p_x )\ge \frac {\hbar}{2}.$$

What is the scientific proof of this principle? Operators Uncertainty

Answer

The uncertainty principle, in the variance formulation, states that in any quantum state $|\rangle$, the quantity

$$\langle (p-

)^2 \rangle \langle (x-\langle x\rangle)^2\rangle \ge {\hbar^2 \over 4} $$

To understand why shifting p and x by their expected value and squaring gives the squared uncertainty, see this answer.

The proof is by noting the following

$$ |\langle \psi | \eta \rangle| \le \sqrt{ ||\psi||^2 ||\eta||^2}$$

This is the statement that the dot-product of two vectors is less than the product of their lengths. It is called the "Cauchy Schwartz inequality". For the special case above, defining the operators $P= p-\langle p\rangle$ and $Q=x-\langle x\rangle$ (and squaring both sides),

$$ ( \langle P Q \rangle )^2 \le \langle PP\rangle\langle QQ\rangle $$

Where to see that the above is an instance of Cauchy Schwarz, take:

$$ |\psi\rangle = P|\rangle$$ $$ |\eta\rangle = Q|\rangle$$ While the product PQ can be decomposed into a real and imaginary part

$$ PQ = {1\over 2} (PQ+QP) + {1\over 2} (PQ-QP) $$

The first part is imaginary, because if you take the Hermitian conjugate, it changes sign. The second part is real (this is ultimately because P and Q are real, i.e. Hermitian). The expected value of PQ squared is the square of the imaginary and real parts separately

$$ (\langle P Q \rangle)^2 = {1\over 4} (\langle [P,Q]\rangle)^2 + {1\over 4}(\langle PQ+QP)\rangle)^2 $$

Since both square things are positive, this means that the left hand side is bigger than one quarter the square of the commutator. The commutator is unchanged by the shifting,

$$ [P,Q] = [p,x] = \hbar $$

So that

$$ \langle P^2 \rangle \langle Q^2\rangle \ge (\langle PQ \rangle)^2 \ge {1\over 4} (\langle [P,Q] \rangle)^2 = {\hbar^2 \over 4} $$

The proof is usually given in one line, as directly above, where the Cauchy Schwarz step (first inequality), the imaginary/real part decomposition (second inequality) and the shifted canonical commutation relations (last equality) are assumed internalized by the reader.

This proof appears on Wikipedia, it is used in all QM books, but perhaps this explanation is clearer.