All Questions
Tagged with hilbert-spaceschroedinger-equation
273 questions
0votes
1answer
59views
Why does a large crystal ($L >>$ wavelength) imply that we have a discretization of wavenumbers $k_x$?
I'm taking an introduction to quantum mechanics course and I'm confused on this topic. I'm specifically talking about the 1D atomic lattice model (I think it's also called the Kronig-Penney Model). ...
3votes
1answer
203views
In perturbation theory, are there two or three summation terms in the second-order correction to the eigenfunctions?
Context This question is a narrow one and it is specifically related to non-degenerte, time-independent perturbation theory. In working through [1], Sakurai offers in Eq. (5.1.44) that the second-...
- 734
1vote
2answers
102views
Why does the free particle potential in QM only allow scattering states?
Griffiths mentioned in his quantum mechanics book that 'because the free particle potential is zero everywhere, it only allows scattering states'. (Scattering states in here are defined as states of $...
1vote
1answer
117views
Schrödinger states in quantum field theory
In the Schrödinger picture of quantum field theory states are defined as functionals of fields and belong to an abstract Hilbert space $H$. The time evolution of these states are governed by a ...
- 4,500
13votes
2answers
629views
Perturbation to a Dirac delta potential well
I am considering an unperturbed Hamiltonian of the form: $$H_0 \equiv -\dfrac{\hbar^2}{2m} \dfrac{d^2}{dx^2} - \alpha\, \delta(x),$$ which has a single bound state solution given by $$\psi^{(0)}(x) = \...
- 1,177
0votes
1answer
203views
A pedagogical explanation for matrix evolution and Schrödinger equation
What is the mathematical connection between Hermitian evolution from state $|\psi\rangle$ to $\langle\chi|$, $$\langle\chi|H|\psi\rangle$$ as matrix expression, and the time evolution expressed with ...
- 137
0votes
0answers
63views
Time dependent Schrödinger equation for dual state
This might be a stupid question, but how does one derive the TDSE (time-dependent Schrödinger equation) for the dual state $\langle \Psi(t)|$? Here is my try: TDSE: $i\hbar\ \partial_t |\Psi(t)\...
- 199
0votes
3answers
101views
Time evolution of energy eigenstates in the Heisenberg picture
It is said that Heisenberg and Schrödinger pictures are equivalent. The way I understand this is that in the Schrödinger picture the state itself evolves in time, but in the Heisenberg picture the ...
8votes
2answers
646views
Is momentum expectation value always 0 for an eigenfunction?
in which cases the following derivation holds? assume the Hamiltonian $$H = \frac{\hat{p}^2}{2m} + \hat{V}(r)$$ now $$[\hat{r},H] = -i\hbar \hat{p} \ \text{and}\ \langle\phi|\hat{p}|\phi\rangle \...
0votes
2answers
124views
What is the difference between the superposition principle and completeness relation in quantum mechanics? [duplicate]
As far as I know, we say that any wavefunction which is a superposition of the solutions of the Schrödinger equation are also valid solutions. On the other hand, according to completeness relation we ...
- 1,629
1vote
0answers
94views
Hellmann-Feynman theorem and the derivation of the Lippmann-Schwinger equation
When deriving the Lippmann-Schwinger equation, one denotes $$H_\text{free}|\phi\rangle = E|\phi\rangle \tag{1}$$ with $H_\text{free}$ as the free Hamiltonian and $$H|\psi\rangle = E|\psi\rangle \tag{2}...
- 309
0votes
1answer
160views
Time dependent perturbation theory validity and initial condition
This question concerns the validity of the perturbation theory formula that is so commonly found. The section to which I explicitly refer is section 18 of Shankar, 2nd edition (pg 473 on). Per usual ...
- 15
0votes
1answer
53views
Reduced dynamics for the pure state of the system
First consider a closed system. If it is known a priori that the initial state of the system is a pure state $| \phi \rangle$, then the von Neumann equation for the density matrix is $$ \frac{d \rho}{...
- 41
6votes
2answers
475views
Why can non-differentiable solutions to the Schrödinger equation be ignored?
To clarify the question, let's consider the particle in a box (infinite potential $V$ outside [0,1], potential 0 inside [0,1]). (But the problems illustrated here also apply to particles in a non-...
4votes
2answers
736views
Does this double well potential contradict the fact that there is no degeneracy for one-dimensional bound states?
This example is taken from Modern Quantum Mechanics by Sakurai. Consider a symmetric double well potential in one-dimension with a barrier of height $V_0$ and width $a$ at the middle. The eigenstates ...
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