Physics Friday 48

Quantum Mechanics and Momentum
Part 5: Momentum Eigenstates

For any observable with operator , we recall that any state with a definite value for the observable has a wavefunction that is an eigenfunction of the operator, with the value of the observable given by the corresponding eigenvalue.

Now, we have that the momentum operator in one dimension is . What, then, are the states of specific momentum (in the position basis)? We set up the eigenvalue equation:

Now, using our operator, this becomes:

This is a simple differential equation (first-order separable), with solution:

where A is a constant (our constant of integration) and is the wavenumber corresponding to the wavefunction momentum p0 via our de Broglie relation .

Note that for all x, so fixed momentum requires equal probability for all positions (when Δp→0, Δx→∞, as required by the Heisenberg uncertanty principle). This, however, means that the wavefunction is not properly normalizable; pure momentum states represent an ultimately non-physical, idealized limit of real wavefunctions as the uncertanty in momentum shrinks.

Similar to the above arguements, we find in three dimensions that the wavefunction for a momentum is

where .

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