Quantum Questions

Unanswered

* How do you understand the operator for momentum in quantum mechanics?
Does every Hermitian operator correspond to an observable? My quantum professor thinks so, but I don't think he actually has a proof.
Does a finite wave necessarily have to be non-monochromatic in reality, or is that implication just a result of the mathematical analysis?
Why is commutation relations all you need to know the properties of everything in quantum mechanics? Mathematically its ok, but there must be some concepts that we don't yet know that make it more obvious when you take a step back and look at the big picture.
What is the difference between operators in quantum mechanics that correspond to measurements and ones that don't? Measurement operators are always Hermitian and non-measurement operators are usually not Hermitian. Non-measurement operators are often unitary in order to preserve probability, with the notable exception of the raising and lowering operators.
Where does the third postulate of quantum mechanics come from - why is the inner product squared?
Is there a general algorithm for going from a set of commutation relations to an operator algebra that encompasses the simple harmonic oscillator algebra and the angular momentum algebra under one structure?
How do we know that quantum wavefunctions apply to macroscopic objects and not just to one particle at a time?
Why do interaction terms show up in the Hamiltonian even though they never contribute to the energy of a state because they only connect different states?
Why is energy the generator of time translation?
How can we prove that the only solutions to the half SHO are the odd SHO solutions?
Is it true that the state vector is a linear combination of all distinguishable states and not just distinguishable energies? (In reference to Spring 2003 #4)
Is it true that you have to use degenerate perturbation theory if and only if the denominator blows up? For example, if you are dealing with a system with degeneracy, but the state you are perturbing from is non-degenerate, then the denominator doesn't blow up, so does this mean you don't have to use degenerate perturbation theory?
What do the orbitals of positronium look like?

Answered

How can the quantum wavefunction have enough freedom to determine the two completely independent parameters of position and momentum? The position is given by an integral of the magnitude and by Sakurai (2.4.22) the momentum operator is determined by the phase. For a complex function, the magnitude and phase are two independent parameters. The reason this was not obvious at first was that the most simple problems have an expectation value of zero for momentum, so the wave function is pure real.
What is the physical difference between the two states (singlet and triplet) of a pair of spin half particles that both have zero z component of total spin? Answer: The singlet state has zero total spin, which means that the two spins are pointing opposite to each other, whereas the triplet state has total spin 1, which means the spins are aligned, but they are not pointing in the z direction.
Are individual photons considered to be monochromatic? No according to Professor Abers, the uncertainty principle applies to them the same as to a beam of light. They have a spread of energy and are localized in space, but have an arbitrary extent. An individual photon can look like any shape wave packet, gaussian or otherwise.
Where does the minimal coupling rule come from? Shankar explains this classically in section 2.2.
When we do an expectation value of the momentum operator, will the result be the mechanical momentum or the canonical momentum? It is the canonical momentum, which is not the same as the mechanical (real) momentum if there are electromagnetic fields present.