My understanding of the wave function is that it shows the probability of observing something like an electron at different positions in a given moment in time. basically the x axis is position and the y axis is the probability (or can be used to find the probability).

the book I’m reading, in describing the uncertainty principle, shows how a wave function localized at a single point has an immeasurable wavelength, and a wave function with repeating waves and a measureable wavelength has multiple different possible locations.

that makes sense to me, and i understand that the amplitude corresponds to location probability, what I don’t understand is how wavelength shows momentum/speed.

the book references work by de Broglie and his discovery that wavelength correlates to momentum, but that doesn’t seem intuitive or obvious. How can you calculate the probability of measuring a certain speed just by looking at the wave function?

In: 2

The simple reason is just duality between position and momentum, due to the Hamiltonian.

In classical physics, position and momentum are conjugated variable, that is, they satisfies the Euler-Lagrange equation, where the Hamiltonian is taken to be the energy functional on the phase space. In other word, there is a physically meaningful way to “rotate” rate of change of position into rate of change of momentum and vice versa. The rotation work kind of like real and imaginary axis of complex number: positive real goes to upper imaginary, which then goes to negative real, then go to lower imaginary, and finally back to positive real.

(if you know Fourier transform, you know where this is going; after all, Fourier transform also perform that kind of rotation)

In quantum mechanics, quantities are replaced by self-adjoint operators (or something even more general) on Hilbert space. The operators allow you to give probability amplitudes to possible measurement, while still allow you to perform computations on measurements as usual. If you are interested in why we do it, much research had been done on exactly what axioms are needed to justify us using Hilbert space and self-adjoint operators.

The infinitesimal time translation operator is not self-adjoint since it’s first derivative over time, so to produce a self-adjoint version, you need to multiply by i, times a constant factor. This gives us a self-adjoint Hamiltonian, because the Hamiltonian is for energy, and energy is what’s preserved over time, so they’re diagonalized by the time translation operator, and hence must be a constant multiple of the made-self-adjoint time translation operator. This constant factor measure the rate of change of phase and hence can be derived from experiment (using an interference pattern, for example), and it turned out to be 1/h (where h is the reduced Planck’s constant).

When physicists tried to derive quantum mechanics, they expect that at large scale, classical principle hold. In other word, they expect the Hamiltonian to still have the same formula as before (barring some change of units). Therefore, the Hamiltonian satisfies the same formula, except with the position and momentum quantities getting replaced by their operators. From that, you can derive a canonical commutation relation, which is just the quantum version of Euler-Lagrange equation.

Finally, from the canonical commutation relation, you can see that in a position basis (that is, you’re looking at the wavefunction that describes probability amplitude for position), then the momentum operators looks like a derivative; and vice versa. Thus, you can transform into the momentum basis by diagonalizing the translation operator, that is, performing a Fourier transform.