- #1

fog37

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My understanding is that the state of the system is ##|\Psi>##. We can take the inner product between the state ##|\Psi>## and the eigenstates of the position operator ##\hat{x}##:

$$<x|\Psi>=\Psi(x)$$

The function ##\Psi(x)## is the wave function we are initially introduced to in beginner quantum mechanics. We call this the position representation of the state ##|\Psi>##. In position space, the eigenfunctions of the position operator ##\hat{x}## are delta functions ##\delta(x-x_0)## located at ##x_0##.

We could also take the inner product between the state ##|\Psi>## and the eigenstates of the momentum operator ##\hat{p_x}## to get a new function:

$$<p|\Psi>=\Psi(p)$$

It turns out that ##\Psi(p)## is the Fourier transform of ##\Psi(x)##. The eigenfunctions of the momentum operator in momentum space are delta functions ##\delta(p-p_0)## located at ##p_0##, correct?

What function would be get by taking inner product between the state ##|\Psi>## and the eigenstates of the energy operator ##\hat{H}## also called Hamiltonian:

$$<H|\Psi>=\Psi(E)?$$

Would we get a function like ##\Psi(E)##?

In which circumstance could we get a function like ##\Psi(x, p_y, E)##? What type of inner product would we need to calculated between the the operators ##\hat{x}##, ##\hat{p_y}##, ##\hat{H}## and ##\Psi(x)##? Would it be possible if the operators were pairwise commuting with each others?