## Physics Friday 49

Quantum Mechanics and Momentum
Part 6: Time, Frequency, and Schrödinger’s Equation

Note that previously, we combined the deBroglie relation between momentum and (angular) wavenumber, the Fourier transform conversion between the position and wavenumber spaces, and the formula for the Fourier transform of a derivative to demonstrate (non-rigorously) that the momentum operator in the position basis is  for a one-dimensional position space, and , where  is here the gradient operator (see here), for a three-dimensional space.

Now, recall that just as we can perform a Fourier transform to change from position to the corresponding wavenumber space, a Fourier transform applied to a function of time returns a function of (angular) frequency:


(introduced here). We also have the deBroglie relation between energy and angular frequency E=ℏω. Thus, we should then expect some form of relation between the time derivative and the operator corresponding to energy; and there is. With  the Hamiltonian operator, which is the operator for energy, we have for any time-varying wavefunction  the relation . This is, in fact, the time-dependent Schrödinger equation. As the Hamiltonian is a linear operator with regards to the wavefunction, we see that the time-dependent Schrödinger equation is a linear differential equation, and that any linear combination of solutions is also a solution.

We see that the time-dependent Schrödinger equation is easily solved for wavefunctions of definite energy; from the nature of quantum operators for obervables, a wavefunction with definite energy En is an eigenfunction of the Hamiltonian with eigenvalue En. Then the time-dependent Schrödinger equation gives:
.
This is a basic separable first-order differential equation (in time), giving solution
, where ψn(x) is the spatial component of the wavefunction (which must be an eigenfunction of the Hamiltonian with eigenvalue En), and ωn is the (angular) frequency corresponding to the energy En via the deBroglie relation E=ℏω.
(Note that the eigenvalue relation  is the time-independent Schrödinger equation.)