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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Constructing solutions to the time-dependent Schrödinger's equation

The following question is from David Griffiths' Introduction to Quantum Mechanics: Problem 2.13 A particle in the harmonic oscillator potential starts out in the state $$\Psi(x,0) = A[3 ...
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1answer
56 views

Solution of the Radial Part of the Schroedinger Equation [closed]

The general Schroedinger Equation is: $$\left[-\frac{\hbar^2}{2m}\triangle +V(r,\vartheta,\varphi)\right]\psi_{nlm}=E\psi_{nlm}$$ When considering free waves, i.e. $V(r,\vartheta,\varphi)=0$ and a ...
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1answer
47 views

Property of the wave functions of a free particle

How can I show that the following holds? $$\langle nlm\mid \partial_z^2\mid nlm\rangle=-\int_0^{4\pi}d\Omega\int_0^{\infty}drr^2\left|\partial_z\psi_{nlm}\right|^2$$ The wave functions of a free ...
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1answer
66 views

Non-relativistic limit of complex scalar field Lagrangian

I am trying to derive the non-relativistic Lagrangian for a complex scalar field from taking the non-relativistic limit of the complex scalar field Lagrangian. I am following the steps in "QFT for ...
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1answer
40 views

Analytical non-separable solution for schrodinger equation

I am an undergraduate with the background of a first course in Quantum Mechanics. I want to find out if there exist non-unique solutions to Schrodinger equation. So I have to find potentials ...
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1answer
60 views

The Tunnelling Man [closed]

What's the probability that I will tunnel through a solid wall? By "tunnel" I mean, the probability of finding me on the other side of the wall. Assumptions Wall thickness = $d$ Clearly state any ...
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0answers
56 views

Selecting physical solutions in numerical eigenvalue problems

I try to solve a certain time-independent Schrodinger equation numerically, using the method of finite differences. My boundary conditions are such that the finite difference method gives me an ...
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1answer
57 views

Finding the wave function of a quantum harmonic oscillator [duplicate]

How can I find the wave function of a quantum harmonic oscillator? If I measure its energy several times, my measurements will change the state of a system. All I know are the possible states, given ...
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1answer
39 views

Particle in a box - speed probability distribution

Consider a particle in a box with infinite barriers. By solving the Schrödinger we can find the probability of finding the particle at some points in the box. How can we find the probability of ...
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1answer
76 views

Obtaining quantum Hamiltonian for charged particle from path integral formulation

I was working on Shankar 8.6.4, which is about obtaining in one dimension the Hamiltonian operator of a charged particle from the path integral formulation. First, I get the propagator over a time ...
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0answers
24 views

How can we derive the schrodinger wave equation? [duplicate]

How can we derive the Schrodinger wave equation with simplified explanation?
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1answer
123 views

Why “textbook examples” of solutions to Schrodinger equation only deal with electrons?

Whenever studying first courses of quantum mechanics, the Schrodinger eqaution is always illustrated by an electron in some kind of a potential, and the solution (wavefunction) represents probability. ...
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100 views

How did Schrödinger come up with his equation?

So all of the people who studied QM know the famous Schrödinger equation. I have read that it was not derived, but it is a postulate; something that is just real. Some people have tried to explain ...
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55 views

Intuition behind the solutions of the Schrodinger equations

I am learning the basic and easy first examples of the most common problems in Quantum Mechanics,and while trying to find solution to the Schrodinger equation,i find myself struggling with the ...
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0answers
44 views

Potential in Schrödinger equation when doing a Galilean transformation

I was looking at the quantum mechanics book by Bransden and Joachain, specifically at the section about Galilean transformations, and I was trying to find out what they did here for the potential ...
2
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2answers
108 views

Time evolution of a wavepacket

I do not understand why if $H\psi = E\psi$, then the time-evolution of the wavefunction is given by $e^{-iEt/h}\psi(x)$.
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Time dependent solution to infinite well

A particle of mass $m$ is confined within an infinite, one-dimensional potential well, $U(x)$, of width $a$. $$ U\left(x\right) = \left\{ \begin{array}{lr} \infty &\: x \leq ...
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3answers
52 views

Calculating the probability of a given energy

Given a normalised wavefunction say $$\psi(x) = A\sin(n\pi x),$$ (where $A$ is a normalisation constant) I can calculate the probability of finding the particle being between a position $x$ and $x + ...
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4answers
172 views

Physical reason why the derivative of a wavefunction has to be continuous?

Question What is the physical reason (i.e. without any maths) that the derivative of a wavefunction (except with infinite potentials) has to be continuous? Other info I know that in the classical ...
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2answers
48 views

Potential step and exponential decay?

Let us say we have a wave going from a region ($x<0$) where the potential is $U_1$ to a region ($x>0$) where the potential is $U_2$. The wave function in the second region takes the form: ...
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66 views

Naive quantization of Schrödinger field

I just started learning QFT and I was wondering if one is able to quantize the Schrödinger field similar to the way one is able to quantize electromagnetic or elastic mechanical wave modes. E.g. ...
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23 views

Metastable $E=0$ s-wave bound state in a spherical potential well

I am currently dealing with scattering theory. I looked up the scattering on a spherical well potential. $$V(r) = \begin{cases} -V_0 & , r \leq R\\ 0 & ,r > R \end{cases} $$ where ...
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1answer
68 views

Why is the ground state energy of a 2DEG higher compared to the 3DEG?

I am reading something about a 2DEG (2-dimensional electrongas model) and can not understand it. My book says the ground state of the 2DEG is higher compared to a 3DEG because the confinement to 2D ...
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1answer
51 views

What does it mean for a quantum particle to have energy $E_n$? And what is its general normalised state?

In this particular case, I have found the energy to be quantised with energy levels $\frac{h^2n^2}{2m} >0 $ where $n$ is an integer. Suppose a particle has energy $E=\frac{4h^2}{2m}$, then this ...
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1answer
62 views

Boundary conditions of the radial Schrodinger equation

Consider the radial differential equation $$\bigg( - \frac{d^2}{dr^2} + \frac{(\ell+\frac{d-3}{2})(\ell+\frac{d-1}{2})}{r^2} + V(r) + m^2 \bigg) \phi_\ell (r) = \lambda\ \phi_\ell (r),$$ which I've ...
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1answer
38 views

Normalizing a wave function in a mixed well

So I got this potential and want to solve for the even wavefunctions http://imgur.com/GKAy4nD Since it's symmetric around the origin I need only to look at the interval [0,b] and solve for the ...
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Ground State energy as a function of $N$ and $B$, $E_0(N,B)$

The one-particle Hamiltonian is given by: $$\hat{H}=\frac{1}{2m}\left(p+\frac{e}{c}A\right)^2$$ where $p=\hbar\vec{k}$ with $e > 0$ and vector potential $A=(0,x,0)B$, such $B=\triangledown ...
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Enforcing the exchange criteria for two particles in a box in different states

Suppose you have two identical particles (for simplicity we can think of spin 0 bosons for which are represented as a scalar wave-functions, but fermions have a similar problem) in a 1D box that ...
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1answer
51 views

Applying Schrodinger equation to find the energies of a free electron model in a metal [closed]

The one-particle Hamiltonian is given by $$\hat{H}=\frac{1}{2m}\left(p+\frac{e}{c}A\right)$$ with $e > 0$ and vector potential $A=(0,x,0)B$, such $B=\triangledown \times A=(0,0,B)$ Question: "I ...
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1answer
42 views

How to determine if a potential admits bound states?

According to Griffith's Quantum Mechanics, "$E$ must exceed the minimum value of $V(x)$, for every normalizable solution to the time independent Schroedinger equation" As an example, there is no ...
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2answers
97 views

What state does the particle in a box occupy?

My textbook derives the equations for the different energy states $E_n$ of the particle in a box. But my professor in class said this example was a good one because it spoke about the "superposition ...
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81 views

Spin in Schrodingers Equations

With the usage of Dirac notation we've gotten around a large amount of of inconvenience that would be dealing with spin. But, I was wondering, do we merely do this because it is inconvenient to try ...
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Is there a physical interpretation of Neumann boundary conditions for the free Schrodinger equation on a domain?

Let $\Omega$ be a domain in $\mathbb{R}^n$. Consider the time-independent free Schrodinger equation $\Delta \psi = E\psi$. Solutions subject to Dirichlet boundary conditions can be physically ...
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WKB boundary conditions [duplicate]

Technically, this is a math problem but I think it is better here than in the math stacks. Consider the differential equation \begin{equation} y'' = (x^4 - E)y \end{equation} The boundary ...
4
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2answers
73 views

Why do we not require higher derivatives to match at boundary when solving the Schrödinger equation in a given potential?

When solving the time independent Schrödinger equation for a given potential in 1D, the main part of the solving involves matching boundary conditions. Usually, we require the value and the first ...
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1answer
75 views

Why is probabilty conserved under time evolution of a system in quantum mechanics?

I've studied quantum mechanics to a certain degree, but one question that I've never been able to get a fully satisfactory answer to is why probability is conserved (by this I mean that it has either ...
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1answer
36 views

Use of termination in solving quantum harmonic oscillator, hydrogen atom etc

I can't seem to understand the use of termination to make the series solutions physically acceptable (when solving the linear harmonic oscillator etc.). So what if the series does not terminate, it's ...
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1answer
84 views

Why is the position space free particle wavefunction a function of momentum?

This is one of those little things that has always confused me. If someone said to you "in quantum mechanics, the eigenfunctions of a free particle are $\exp(ipx/\hbar)$" how would you know that ...
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1answer
44 views

Infinite Energies of a particle in a rectangular box

For a particle trapped inside a rectangular box of side lengths $l_x$ $l_y$ and $l_z$, the energies are ...
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0answers
61 views

Unitary evolution operator

Assume we have a system in a state $\psi$ that is a superposition of eigenvectors of some observable $A$. Now we make a measurement of the observable $A$; the state after the measurement $\phi$ is a ...
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2answers
96 views

Time dependent Hamiltonian and Gauge invariance

In general, in quantum mechanics we can prove probability current or the Schrodinger equation and other quantities are gauge invariant. However, the Hamiltonian isn't gauge invariant. Under a gauge ...
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1answer
96 views

Probability of finding particle in infinite square well, displaced walls

Initially a quantum particle moves in a one-dimensional well ($x$-axis) from $-a$ to $ a$, $ V = \infty $ outside and $ V = 0 $ inside the well. So initially, the wave-function $$ \psi_0 = ...
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2answers
65 views

Importance of bound states

While solving a potential well problem we get scattering states and bound states (if exist). Number of the bound states we get depends on the potential profile. What I want to ask is, what is the ...
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0answers
29 views

How do I find the electron confinement energies in a spherical quantum dot?

So if I've got a spherical quantum dot, we'll say it has a 10nm diameter for simplicity. This dot is a semiconductor and it has an electron with an effective mass altered by a factor of 0.2. How do I ...
5
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1answer
85 views

Has anyone published the procedure to generalize ladder operators for any potential in Schrodinger's equation?

I know that the ladder operator for the quantum harmonic oscillator \begin{align} H\psi_m = \left(\dfrac{p^2}{2m}+\dfrac{1}{2}m\omega^2x^2\right)\psi_m=E_m\psi_m \end{align} is \begin{align} A = ...
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4answers
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Why aren't orbitals symmetric?

In an hydrogen-like atoms the orbitals are solutions to the Schrodinger equation suitable for the problem. They describe the regions where an electron can be found. So, why don't they have spherical ...
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2answers
72 views

Particle in a one dimensional box conditions

Why does the wave function have to be $C^1(\mathbb{R})$ for a finite square well but not for an infinite square well? For an infinite square well with boundaries at $x=0$ and $x=L$, we have ...
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3answers
270 views

Why does the density matrix $\rho$ obey a wrong-signed Heisenberg equation of motion?

The density matrix is defined as $$ \rho_\psi ~:=~ \frac{\lvert\psi(t)\rangle \langle \psi(t)\vert}{ \langle \psi(t) |\psi(t)\rangle }$$ in the Schrödinger picture. $\rho_\psi$ is obviously a time ...
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1answer
119 views

Finding finite square well width and depth from transmission resonance

For an electron incident a one-dimensional finite square well the transmission probability is $\approx$1 for electron energies $E_1=0.6 \textbf{ eV}$, $E_2=1.9 \textbf{ eV}$ and $E_3=3.4 \textbf{ ...
6
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1answer
366 views

What is a 'turning point' in WKB and why does it fail at that point?

What is meant by a classical turning point in quantum mechanics and why does the WKB approximation fail at that point?