Linked Questions

17
votes
4answers
13k views

Why do we use Hermitian operators in QM?

Position, momentum, energy and other observables yield real-valued measurements. The Hilbert-space formalism accounts for this physical fact by associating observables with Hermitian ('self-adjoint') ...
13
votes
4answers
4k views

Intuitive meaning of the exponential form of an unitary operator in Quantum Mechanics

I'm an undergraduate student in Chemistry currently studying quantum mechanics and I have a problem with unitary transformations. Here in my book, it is stated that Every unitary operator $\hat{\...
6
votes
2answers
3k views

Can expectation value be imaginary?

I was solving a problem and the result of the expectation value of an operator came out to be $-\frac{\hbar}{4}$ $i$. Is this result possible? It seems counter intuitive.
2
votes
1answer
5k views

How do I show that the eigenstates of a Hamiltonian can be made orthonormal?

I've been tearing my hair out over this all evening. It should be simple but I must be missing something somewhere. Can someone show me how to prove that the eigenstates of a Hamiltonian can be made ...
5
votes
1answer
2k views

Non-Hermitian operator with real eigenvalues?

So we know that in Quantum Mechanics we require the operators to be Hermitian, so that their eigenvalues are real ($\in \mathbb{R}$) because they correspond to observables. What about a non-Hermitian ...
0
votes
4answers
1k views

Schrödinger equation and non-Hermitian Hamiltonians

Is the Schrödinger equation still valid if we use a non-Hermitian Hamiltonian with it? By this I mean does: $$\hat{H}\psi(t) = i\hbar\frac{\partial}{\partial t}\psi(t)$$ if $\hat{H}$ is not ...
2
votes
2answers
852 views

Is there any non-hermitian operator on Hilbert Space with all real eigenvalues?

The property of hermitian is the sufficient condition for eigenvalue being real. Is there any non-hermitian operator on Hilbert Space with all real eigenvalues? If there exist, then can all ...
2
votes
2answers
365 views

Expectation value of an imaginary operator acting on a real function

In a video (http://youtu.be/r_gBQ_qhg8U?t=9m58s) it's stated that a matrix element of an imaginary operator acting on a real wave function is zero, i.e. $$\langle\text{real}|\text{imaginary}|\text{...
1
vote
1answer
400 views

Hermiticity of the quantum field

The quantum field resultant from the quantization of a real classical field is hermitian, but why the quantum field corresponding to a complex classical field should be non-hermitian?
2
votes
2answers
277 views

Average momentum in quantum mechanics over some finite interval of space

Why can't the expectation value of momentum be computed over some finite interval of space? Something like, $$ \int_a^b \psi^* \hat{p}\psi ~\mathrm{d}x.\tag{1}$$ I understand that usually we compute ...