Is the classical relationship between energy and momentum valid in quantum physics? Here we are talking about non-relativistic quantum  physics. So we all know kinetic energy $T = E - V = \frac{1}{2}mv^2$ in classical physics. Here $V$ is the potential energy of the particle and $E$ is the total energy. Now what I am seeing is that this exact same relation is being used in quantum physics with impunity. For example when it comes to a free particle, this relation was used to get relation between its wave number and angular frequency.
From what I understand, this classical relation still holds, but the only thing new here is that $E$ and $p$ are not known precisely but instead we have a statistical distribution for them and so therefore even though the classical relation holds we cannot use it (because it requires exact values to be fed into it and not expectation values). Am I getting it right? \
But then again we know that for a stationary state energy $E$ is fixed and so by above reasoning $p$ must always be fixed for a stationary state as well. This doesn’t sound right. So what am I missing?
 A: I am not quite sure what you mean. It is certainly true in QM that the Hamiltonian is equal to the kinetic plus potential part, $H=T+V$. So $T=H-V$. The energy is the expectation value (average measurement value) of the Hamiltonian. So $\langle T \rangle = \langle H\rangle-\langle V\rangle=E-\langle V \rangle$. So the relation holds between the averages.
If we are at an energy eigenstate, then we will always get the same result if we measure the energy. This does not generally imply that we will get the same result if we measure the momentum. In some cases, notably a free particle, the energy eigenstates are the momentum eigenstates so we will indeed always get the same momentum. In others, notably a bound electron in an atom, we would get a probability distribution of momentum measurements (for a single energy eigenstate).
A: In classical mechanics, the energy of an mechanical oscillator for example is the sum of its elastic and kinetic energy, that are function of position and momentum respectively.
In QM, the hamiltonian operator is the sum of potential and kinetic energy, that are function of the operators position and momentum respectively. The relations are between operators, not values.
The eingenvalues of the operators have probabilities according to the state of the system. If it is an eigenstate (of energy for example), the eingenvalue is known.
