# Tagged Questions

Partial differential equation which describes the time evolution of the wavefunction of a quantum system. It is one of the first and most fundamental equations of quantum mechanics.

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### Free particle Schrodinger equation: propagator

I am going through Shankar's Principles of Quantum Mechanics and am having trouble finding the free particle propagator $U(t)$ that satisfies $\lvert\psi (t)\rangle = U(t)\lvert\psi (0)\rangle$ due to ...
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### Minimum gap between consecutive energy levels?

Assume a standard one-particle, non-relativistic Hamiltonian of the form $$H=\frac{p^2}{2m}+V(r)$$ and denote its eigenvalues as $E_{n,\tau}$, where $n$ is the principal ...
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### Degeneracy of energy levels of a particle in a spherical step potential?

I have a particle of mass $m$ and spin $1/2$, in a spherical step potential, $$V(r) = \begin{cases} 0 & r<a, \\ V_0 & r>a. \end{cases}$$ Now they ask me to find, without solving the ...
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### Problems on rectangular well [closed]

We have a rectangular potential well from $x=-b$ to $x=a$.Can we divide the well into two parts from $x=-b$ to $0$ and $x=0$ to $a$ to solve Schrodinger equation?Actually,I want to know how to solve ...
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### Quantization of energy in semi-infinite well

Consider an electron with total energy $E>V_2$ in a potential with $$V(x)= \begin{cases} \infty & x< 0 \\ V_1 & 0< x< L \\ V_2 & x>L \end{cases}$$ ...
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### Existence of eigenstates in type III staggered semiconductor heterojunctions?

Two semiconductors are aligned in the type III staggered fashion and sandwiched in between infinite potentials. Can there be an eigenstate across the valence band of one material and the conduction ...
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### Protocol for solving time independent Schrodinger equation

Just a short question about the protocol for solving the time-independent Schrodinger equation for different potentials and the reasons for accepting and rejecting solutions. Take for example the ...
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### Gauge transformation of vector potential multiplies wavefunction by phase

Consider an electron in an electromagnetic field with scalar and vector potentials $\phi, \mathbf{A}$. Suppose for simplicity that $\mathbf{A}$ is time independent. Suppose also that we know the ...
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### What factors relate the number of protons in the nucleons with the number of electrons in the orbitals?

Atoms always want to have a closed shell, because it requires low energy compared to the lattice enthalpy. How does this always match throughout the periodic table between the number of protons and ...
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### Purpose of the multidimensional NLSE/GNLSE

I know the purpose of the NLSE (Evolution of a complex field envelope in a nonlinear dispersive medium). Usually I am solving the 1d-GNLSE when simulating the propagation of a light pulse through a ...
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### Is there any Hamiltonian that contains time derivative? [duplicate]

Quantum mechanics is governed by Schrodinger's equation: $$\hat{H}\psi=i\hbar\partial_t \psi$$ It seems that Hamiltonian acts on wave functions like a time derivative. Just out of curiosity, is ...
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### Finding the velocity of a given wavepacket [closed]

I've been given a wave packet, that is moving from right to left toward a (known) potential, which has in time $t = 0$ has the form: $$ψ(x, t = 0) = Ae^{−c(x−x_0)^2}e^{ik_0x}$$ and I need to ...
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### Finding the initial state in the power method for Hamiltonian diagonalization

In section III of the lecture note Chapter 1: Exact Diagonalization, Weimer has described the Power method for Hamiltonian diagonalization. The process requires the choice of an random initial state ...
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### Schrödinger's interpretation of his wave function before Born

The below shows some excerpt from Feynman's lecture notes. 21–4 The meaning of the wave function When Schrödinger first discovered his equation he discovered the conservation law of Eq. (21....
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### Can Schrödinger Equation be derived from Huygens' Principle?

Notes of Enrico Fermi start from an analogy between mechanics and optics and with 4 pages he derives the Schrödinger equation. In all my courses, I have seen as an axiom - this is how wave-particles ...
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### Energy of central potential in QM

A hydrogen atom (Coulomb potential) has energy that only depends on $n$ (if we ignore other effects like spin-orbit coupling). In general (not necessarily Coulomb, can be any V), does $E$ depend on ...
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### Eigenstates of 2D harmonic oscillator in a constant magnetic field

I want to find the eigenstates of the 2D harmonic oscillator in a constant magnetic field $\vec B = \vec B(x,y)$. My Hamiltonian reads $H_0 = H_{xy} + H_z$ where $H_{xy}$, is the hamiltonian of the ...
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### Propagating a Gaussian wavepacket backwards in time

So, I'm following the MIT OCW lectures on 8.04 quantum mechanics by Prof. Allan Adams. I have the expression for the probability distribution of a gaussian wavepacket for a free particle situation. No ...
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### Rate of the increase of width of a Gaussian wavepacket

So, I'm following the MIT OCW lectures on 8.04 quantum mechanics by Prof. Allan Adams. I have the expression for the probability distribution of a gaussian wavepacket for a free particle situation. No ...
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### Non-separable solutions of the Schroedinger equation

I'm studying an undergraduate Quantum Mechanics course and I have some doubts about the solution of the Schroedinger equation by the separation of variables method. If we suppose that the solutions ...
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### WKB Approximation on an linear + harmonic potential

I have a quick question: I have performed the WKB approximation to find the energies of bound states in symmetric potentials (Square, harmonic, ...). To do this I just find the "turning points" by ...
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### Deriving Rabi oscillations using the Heisenberg picture of QM

The semiclassical treatment of an simple two level atom in a resonant electromagnetic field is usually done in the Schrodinger/Interaction picture of QM, by assuming that the wavefunction of the atom ...
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### Is the existence of a sole particle in an hypothetical infinite empty space explicitly forbidden by QM?

Suppose the universe is completely empty with one sole particle trapped in it. To simplify, I will only be looking at the one dimensional case. However, all arguments are applicable for three ...
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### Why is the energy shift due to a 'sagging' potential negative and independent of box size?

Consider a box of width $L$ and the composed of the following potential $$V(x)=\frac{V_0x(x-L)}{L^2}, x\in[0,L]$$ and $V(x)=\infty$ elsewhere. Using perturbation theory - with a square box as the ...