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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30 views

Minimum gap between consecutive energy levels?

Assume a standard one-particle, non-relativistic Hamiltonian of the form \begin{equation} H=\frac{p^2}{2m}+V(r) \end{equation} and denote its eigenvalues as $E_{n,\tau}$, where $n$ is the principal ...
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28 views

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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42 views

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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1answer
43 views

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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1answer
63 views

Schrödinger's Equation with multi-part potential

I have this potential $$V(x) = \left\{ \begin{array}{ll} \infty & \mbox{if } x < -a \\ \frac{V_o}{a}x & \mbox{if } -a \leq x \leq a \\ V_o & \mbox{if } x \geq a \ \end{...
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20 views

Heuristic derivation of Schrodinger equation [duplicate]

Is there a heuristic derivation of Schrodinger equation beginning with concepts from classical physics? When I say classical I mean something at the level of an engineer.
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48 views

Questions About Quantum Delta Function Potentials [closed]

I didn't think that it would be possible for a wave function to get through the delta function because there is no "leakage" of the wave function through an infinite potential barrier. I can ...
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4answers
679 views

Increasing a potential causes increase in energy levels

Suppose a potential $V(x)$, and suppose a bound particle so the allowed energy levels are discrete. Suppose a second potential $\widetilde{V}(x)$ such that $\widetilde{V}(x) \geq V(x)$ for all $x$ (...
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1answer
21 views

Scattering from a step potential barrier

Suppose a potential barrier of the form $$ V(x) = \begin{cases} V_0 & x>0 \\ 0 & x<0 \end{cases} $$ Then, for energy $E$ such that $E < V_0$, we have that the transmission and ...
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27 views

Anharmonic quantum oscillator with momentum perturbation

Given the following quantum oscillator for a particle with mass $m$, and perturbation $-\gamma P$ ($\gamma$ is a constant): $$H=\frac{P^2}{2m}+\frac{1}{2}m\omega^2X^2-\gamma P$$ One could find the ...
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17 views

Darwin term affecting hydrogen s-states

The Darwin term, a correction to the non-relativistic hydrogen Hamiltonian due to the zitterbewegung of the electron, is given by $$H_{Darwin}=\frac{e^2\hbar^2}{8m^2c^2\epsilon_{0}}\delta^3(\...
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2answers
17 views

Normalization of time-independent Schroedinger equation in Spherical Coordinates

I have a short question about the time-independent Schroedinger equation in Spherical coordinates: $$\psi(r, \theta, \phi) = R(r)Y(\theta, \phi)$$ then the normalization condition becomes $$\int |\...
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1answer
49 views

Infinite square well - periodic boundaries

If we have an infinite square well, I can follow the usual solution in Griffiths but I now want to impose periodic boundary conditions. I have $\psi(x) = A\sin(kx) + B\cos(kx)$ with boundary ...
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1answer
23 views

Neutron in a magnetic field (Schrödinger Equation, Eigenstates, Eigenvalues).

Consider the spin of a neutron in a magnetic field $\vec{B}$. A neutron is a neutral particle with the mass of a proton and the spin $\frac{1}{2}$. The Hamiltonian is $H=\mu_n\vec{S}\cdot\vec{B}$...
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73 views

How to plot numerically the wave functions according to the Hamiltonian?

It is often difficult to analytically solve the Schrodinger equation, and so we need to obtain a solution numerically. An example plot is shown below. Here, the wave functions for a three junction ...
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1answer
168 views

Feynman's derivation of the Schrödinger equation

I'm reading the following article: Feynman's derivation of the Schrödinger equation In this article, the autor claims that Feynman derivation of the Schrödinger equation was a key aspect of the ...
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1answer
32 views

Normalisation of angular wave function: particle in a circular box

For a particle in a circular box (with radius $R$) with zero potential inside the circle and infinitely high potential outside of the circle, the Schrödinger equation in polar coordinates is: $$-\...
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31 views

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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1answer
41 views

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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2answers
188 views

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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1answer
37 views

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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1answer
96 views

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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61 views

Solution for Schrödinger equation for constant box potential?

It is known that in a box potential, when we set $V = 0$ inside and $V = \infty$ on the boundaries, the solution to the equation $$ - \frac{\hbar}{2m} \bigg( \frac{\partial^2}{\partial x^2} + \frac{ \...
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2answers
37 views

Are negative energy eigenstates orthogonal to plane wave states?

Orthogonality in discrete Hilbert spaces is straightforward - those encountered by typical examples of infinite wells of any type, spin systems etc. Continuous Hilbert spaces are fine too - we ...
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84 views

Where does the position operator come from?

In quantum mechanics the momentum and energy operators appear in Schroedinger's equation. In fact in the derivation of Schroedinger's equation from the classical wave equation the momentum operator ...
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49 views

Energy conservation and time translations

The time translation is given by a finite unitary transformation $$\hat{U_{\tau}}(\hat{H}) = e^{\big(\frac{i}{\hbar}\tau \bar{H}\big)}.$$ Where $$\hat{U_{\tau}}(\hat{H})|\psi(t) \rangle = |\psi(t-\tau)...
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1answer
54 views

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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30 views

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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3answers
218 views

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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54 views

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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168 views

Schrodinger equation for time-dependent Hamiltonian?

Can I write a Schrodinger equation for time-dependent Hamiltonian like this: $$i\hbar\frac{d}{dt}\psi(t) = H(t)\psi(t)$$ and then perform Euler integration like this: $$\psi(t+\Delta t) = (1-\frac{...
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67 views

Wavefunction of electron in 3D infinite well with non-zero potential

Consider an electron moving in a potential $V$ defined by $$V(x,y,z) = \left \{ \begin{array}{ll} \alpha(x^2 + y^2) & 0 \leq z \leq a \\ \infty & \text{otherwise} \end{array} \right. $$ ...
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1answer
77 views

Schroedinger and Klein-Gordon equation and their complex conjugate

Let's consider the Schroedinger equation \begin{equation} i\hbar\frac{\partial}{\partial t}\psi=-\frac{\hbar}{2m}\nabla^2\psi \end{equation} If I have a wavefunction $\psi$ as a solution, then its ...
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72 views

Spacetime background of Quantum mechanics [closed]

Why is it said that the Schrodinger equation suggests a fixed, non-dynamical background spacetime, with time as an external parameter? How does this interpretation come about from the Schrodinger ...
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53 views

Representing Hamiltonian in discrete position basis

I am trying to numerically find eigenstates of an Hamiltonian. Let $V(x)$ be some potential. Suppose my space is between $-L,L$ and that the allowed positions are every $\Delta x$, such that $\frac{1}{...
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1answer
24 views

Probability to measure momentum of a certain range (eigenfunctions and such)

At a certain point in time a particle of mass $m$ has the corresponding function (function of $x$) $$\psi(x)=\begin{cases}Nx \exp[-bx]~~&\text{for}& x\geq 0 \\ 0 ~~&\text{for}& x&...
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44 views

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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1answer
50 views

Is the Schrodinger equation true for any operator? [closed]

I am going to use the notation $\hat{u}$ to refer to the operator version of $u$. For energy, E, and position $\hat{x}$. If we have a closed system and energy does not change over time: $$ E \hat{x} ...
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What is the cause of discrete or quantized energy levels in an atom? [duplicate]

I understand how it is that electrons move from one energy state to another, however I've not been able to find anywhere that describes why an atom has any particular states. Why should an atom of ...
10
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1answer
162 views

What is known about the hydrogen atom in $d$ spatial dimensions?

In a first (or second) course on quantum mechanics, everyone learns how to solve the time-independent Schrödinger equation for the energy eigenstates of the hydrogen atom: $$ \left(-\frac{\hbar^2}{2\...
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1answer
74 views

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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1answer
37 views

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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2answers
640 views

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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1answer
53 views

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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1answer
40 views

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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1answer
45 views

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 ...