University of Oxford Mathematician Dr Tom Crawford derives Heisenberg’s Uncertainty Principle in Quantum Mechanics with assistance from Micheal Penn. This is the second video in the short series – part 1 on Lie Algebras is on Michael’s channel here.
The video begins with Schrödinger’s Equation for the quantum wave function. By looking for a separable solution we are able to solve for the time dependent component as a complex exponential. The equation for the remaining spatial component is referred to as the Stationary State Schrödinger Equation. This is then expressed in terms of the momentum and position operators which together make up the Hamiltonian.
The definitions of the expectation and dispersion (standard deviation) of an operator are given in terms of a complex inner product, as these will be required later. We then compute the commutator of the position and momentum operators and see that it is non-zero.
Finally, the key proposition which is used to derive Heisenberg’s Uncertainty Principle is stated and proved. Using the positivity of a quadratic function we conclude that the discriminant must be negative, which gives a result about the expectation of two operators and their commutator. By an appropriate choice of the operators we obtain Heisenberg’s Uncertainty Principle: the product of the error in the measurement of the position multiplied and the error in the measurement of the momentum, must be greater than or equal to a strictly non-zero value.