College Math Teaching

July 25, 2011

Quantum Mechanics and Undergraduate Mathematics VI: Heisenberg Uncertainty Principle

Filed under: advanced mathematics, applied mathematics, physics, probability, quantum mechanics, science — collegemathteaching @ 10:05 pm

Here we use Cauchy-Schwartz inequality, other facts about inner products and basic probability to derive the Heisenberg Uncertainty Principle for incompatible observables A and B . We assume some state vector \psi which has not been given time to evolve between measurements and we will abuse notation by viewing A and B as random variables for their given eigenvalues a_k, b_k given state vector \psi .

What we are after is the following: V(A)V(B) \geq (1/4)|\langle \psi, (AB-BA) \psi \rangle|^2.
When AB-BA = c we get: V(A)V(B) \geq (1/4)|c|^2 which is how it is often stated.

The proof is a bit easier when we make the expected values of A and B equal to zero; we do this by introducing a new linear operator A' = A -E(A) and B' = B - E(B) ; note that (A - E(A))\psi = A\psi - E(A)\psi . The following are routine exercises:
1. A' and B' are Hermitian
2. A'B' - B'A' = AB-BA
3. V(A') = V(A) .

If one is too lazy to work out 3:
V(A') = E((A-E(A))^2) - E(A -E(A)) = E(A^2 - 2AE(A) + E(A)E(A)) = E(A^2) -2E(A)E(A) + (E(A))^2 = V(A)

Now we have everything in place:
\langle \psi, (AB-BA) \psi \rangle = \langle \psi, (A'B'-B'A') \psi \rangle = \langle A'\psi, B' \psi \rangle - \langle B'\psi, A' \psi \rangle = \langle A'\psi, B' \psi \rangle - \overline{\langle A'\psi, B' \psi \rangle} = 2iIm\langle A'\psi, B'\psi \rangle
We now can take the modulus of both sides:
|\langle \psi, (AB-BA)\psi \rangle | = 2 |Im \langle A'\psi, B'\psi \rangle \leq 2|\langle A'\psi, B'\psi\rangle | \leq 2 \sqrt{\langle A'\psi,A'\psi\rangle}\sqrt{\langle B'\psi, B'\psi\rangle} = 2 \sqrt{\langle A\psi,A\psi\rangle}\sqrt{\langle B\psi,B\psi\rangle} = 2\sqrt{V(A)}\sqrt{V(B)}

This means that, unless A and B are compatible observables, there is a lower bound on the product of their standard deviations that cannot be done away with by more careful measurement. It is physically impossible to drive this product to zero. This also means that one of the standard deviations cannot be zero unless the other is infinite.


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