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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Discreteness of set of energy eigenvalues

Given some potential $V$, we have the eigenvalue problem $$ -\frac{\hbar^2}{2m}\Delta \psi + V\psi = E\psi $$ with the boundary condition $$ \lim_{|x|\rightarrow \infty} \psi(x) = 0 $$ If we ...
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1answer
43 views

Time evolution of a quantum state

I have another point in QM that I would like clarified. Suppose $$\{|n\rangle\}$$ is a set of eigenstates of both the Hamiltonian $H$ and another operator $\hat O$ corresponding to an observable also. ...
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1answer
98 views

Differential equation (Greens function) satisfied by the kernel using path integrals

I'm reading Feynman and Hibbs, Quantum Mechanics and Path Integrals. How do I show that the kernel $$\tag{2-25} K(x_2 t_2;x_1 t_1)=\int e^{\frac{i}{\hbar}S[2,1]}\mathcal{D}x$$ satisfies the ...
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What values should the solved time-independent Schrodinger equation return? [closed]

I'm doing a project on Schrodinger's equation for my differential equations class. We solved the time independent function, and now we want to provide some examples of applying the equation by solving ...
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1answer
63 views

Periodic boundary condition on a Wave Function of a Particle in a Box

Until now solving the Schrodinger Equation for a particle in a box was relatively easy because the boundaries conditions imposed zero value on the wave function at the boundaries. But now I must find ...
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1answer
122 views

Change of basis in non-linear Schrodinger equation

At the mean-field level, the dynamics of a polariton condensate can be described by a type of nonlinear Schrodinger equation (Gross-Pitaevskii-type), for a classical (complex-number) wavefunction ...
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45 views

Time evolution of Gaussian wave packet

I'm slightly confused as to answer this question, someone please help: Consider a free particle in one dimension, described by the initial wave function $$\psi(x,0) = ...
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57 views

Electron in an infinite potential well

Does this problem have any sense? Suppose an electron in an infinite well of length $0.5nm$. The state of the system is the superposition of the ground state and the first excited state. Find the ...
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55 views

The gauge-invariance of the probability current

It is simple to show that under the gauge transformation $$\begin{cases}\vec A\to\vec A+\nabla\chi\\ \phi\to\phi-\frac{\partial \chi}{\partial t}\\ \psi\to \psi ...
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2answers
89 views

Do we always ignore zero energy solutions to the (one dimensional) Schrödinger equation?

When we solve the Schrödinger equation on an infinite domain with a given potential $U$, much of the time the lowest possible energy for a solution corresponds to a non-zero energy. For example, for ...
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1answer
42 views

Time Dependent HydroHow would I go about writing the time dependent wave function given the wavefunction at $t=0$? gen Wave Function

1) How vwoulHow would I go about writing the time dependent wave function given the wavefunction at $t=0$? go about writing the time dependent wave function given the wavefunction at $t=0$? ...
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3answers
96 views

How do I determine the location of a free particle with Schrödinger's equation?

I'm trying to get to grips with the Schrödinger equation by looking at a free particle. I'm certain at some point I massively misunderstood something. According to a textbook and a lecture the free ...
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1answer
138 views

Schrödinger equation for a harmonic oscillator

I have came across this equation for quantum harmonic oscillator $$ W \psi = - \frac{\hbar^2}{2m}\frac{d^2\psi}{dx^2} + \frac{1}{2} m \omega^2 x^2 \psi $$ which is often remodelled by defining a new ...
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1answer
178 views

What type of PDE are Navier-Stokes equations, and Schrödinger equation?

What type of PDE are Navier-Stokes equations, and Schrödinger equation? I mean, are they parabolic, hyperbolic, elliptic PDEs?
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3answers
107 views

When does the time independent Schrödinger equation have a physical solution?

In some cases, such as finite and infinite square wells, the Hamiltonian has energy eigenstates which correspond to physical wavefunctions. In other cases, such as a one dimensional universe with ...
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1answer
73 views

Tunneling and transmission

Lets say we have a tunelling problem in the picture, where $W_p$ is a finite potential step: If particle is comming from the left a general solutions to the Schrödinger equations for sepparate ...
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2answers
188 views

Why must the angular part of the Schrodinger Equation be an eigenfunction of L^2?

I was reading about the solution to the Schrodinger Equation in spherical coordinates with a radially symmetric potential, $V(r)$, and the book split the wavefunction into two parts: an angular part ...
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2answers
93 views

Solve the angular part of Schrodinger equation numerically

I would like to solve the angular part (the one for what is usually called the $\theta$ angle) of a time-independent 3D Schrodinger equation $$ \frac{\mathrm{d}}{\mathrm{d}x}\left[ (1-x^2) ...
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1answer
86 views

The Klein–Gordon equation

As we know that the Schrödinger equation presents basis of Quantum Mechanics and analogy with Newton second law in Classical Mechanics, I thought that relativistic interpretation of Schrödinger ...
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1answer
98 views

Potential step and its transmission / reflection

Lets say we have a potential step with regions 1 with zero potential $W_p\!=\!0$ (this is a free particle) and region 2 with potential $W_p$. Wave functions in this case are: \begin{align} ...
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2answers
97 views

Why the hydrogen radial wave function is real?

Why the hydrogen radial wave function is real? Is it a coincidence?
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1answer
133 views

How does a state in quantum mechanics evolve?

I have a question about the time evolution of a state in quantum mechanics. The time-dependent Schrodinger equation is given as $$ i\hbar\frac{d}{dt}|\psi(t)\rangle = H|\psi(t)\rangle $$ I am ...
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43 views

Wave equations for two intervals at Potential step

Lets say we have a potential step as in the picture: In the region I there is a free particle with a wavefunction $\psi_I$ while in the region II the wave function will be $\psi_{II}$. Let me ...
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1answer
100 views

Energies and numbers of bound states in finite potential well

Hello I understand how to approach finite potential well (I learned a lot in my other topic here). However i am disturbed by equation which describes number of states $N$ for a finite potential well ( ...
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2answers
285 views

Plotting $\psi$ for finite square well potential

Lets say we have a finite square potential well like below: This well has a $\psi$ which we can combine with $\psi_I$, $\psi_{II}$ and $\psi_{III}$. I have been playing around and got expressions ...
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1answer
109 views

Finite potential well - transcendent equation for even solutions

I have a finite square well like the one on the picture below: I have done some calculations on it and got a transcendental equation for even solutions which is like this: $$ ...
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39 views

exponential potential quantization

What are the energies $E_{n}$ of the Schroedinger operator $$ -\frac{d^{2}}{dx^{2}}y(x)+ae^{bx}y(x)=E_{n}y(x) $$ for some real and positive 'a' and 'b' with the Boundary conditions $ ...
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63 views

Potential step solution is not normalizable

If you try to integrate the solution of the potential step, http://en.wikipedia.org/wiki/Solution_of_Schr%C3%B6dinger_equation_for_a_step_potential#Solution you will notice that it diverges! Doesn't ...
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1answer
146 views

Finite, square, potential well

Lets say we have a finite square well symetric around $y$ axis (picture below). I know how and why general solutions to the second order ODE (stationary Schrödinger equation) are as follows for ...
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2answers
88 views

Finite square well

I know how and why we use this form of stationary Schrödinger equation for finding $\psi$ outside the finite square potential well: $$\frac{d^2 \psi}{dx^2}=\kappa^2 \psi$$ I Also know that the ...
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92 views

Scattering and partial wave analysis for cross section [closed]

Problem Given the central potential: $V(r)=-\frac{\hbar^2}{m a^2}\frac{1}{\cosh({r\over a})}$ and given that we know the solution to the following ODE $\frac{d^2 y}{dx^2}+k^2 ...
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1answer
90 views

Is normalization consistent with Schrodinger's Equation?

Schrodinger's Equation does not set a limit on the size of wave functions but to normalize a wave function a limit must be set. How is this consistent physically and mathematically with Schrodinger's ...
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343 views

Quantum Mechanics: Show that the expectation value of angular momentum does not change with time

The potential is given by $V\left(\left\|(x,y,z)\right\|\right)$, so $[\hat{L}, \hat{H}] = 0$. Using the definition of $\langle \hat{L} \rangle$ and the time-dependent Schrödinger equation, show that ...
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49 views

Question about the linearity of wave functions

For piece-wise constant potential, the potential energy is constant so the time dependent wave function can take the form $\psi(x,t)=C_1e^{i(kx- \omega t)}+C_2e^{i(-kx-\omega t)}$ where ...
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284 views

Galilean invariance of the Schrodinger equation

I am only asking this question so that I can write an answer myself with the content found here: http://en.wikipedia.org/wiki/User:Likebox/Schrodinger#Galilean_invariance and here: ...
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2answers
156 views

How would Lagrangian be used tor recover Schrodinger equation?

In path integral formulation of quantum mechanics, I heard that Lagrangian is defined. So, how would Lagrangian in this formulation be used to recover Schrodinger equation that we normally use?
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1answer
92 views

How to tell if a complex exponential blows up

I'm following Griffiths' Introduction to Quantum Mechanics, where he's discussing the general solution to the delta-function potential problem. The solution he refers to is ...
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1answer
137 views

Usage of Schrödinger equation vs Madelung equations

It is well known that Madelung formulation is alternative to the Schrödinger Formulation, cf. this previous Madelung transformation Phys.SE post. I wanted to know what makes Schrödinger's formulation ...
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1answer
96 views

Constant-dependent potential in radial Schrodinger equation

Studying quantum mechanics, I've found an exercise I don't know how to solve it. Given the radial Schrödinger equation, $$\left [ \frac{d^2}{dr^2}+k^2-\frac{2m}{\hbar^2}\lambda U\left ( r \right ) ...
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1answer
179 views

Solving the 1-D Schrodinger equation for a free particle: Confused about 2 possible general solutions

I am following Griffiths' Introduction to Quantum Mechanics, as well as an online lecture that follows a different book, and both sources give different equations for the general solution of the 1-D ...
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1answer
385 views

Solving the time independent Schrodinger equation: Does a complex solution make sense?

In my notes, I have the Time Independent Schrodinger equation for a free particle $$\frac{\partial^2 \psi}{\partial x^2}+\frac{p^2}{\hbar^2}\psi=0\tag1$$ The solution to this is given, in my notes, ...
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2answers
467 views

Particle in infinite potential well which is doubled in size at $t_0$

I am currently studying for an exam in Quantum Mechanics and came across a solution to a problem I have trouble with understanding. The Problem: A Particle sits in an infinite potential well ...
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0answers
49 views

Proving One-dimensional Wave with Classic Expression [closed]

A general form of a one-dimensional wave is ψ(x) = sin kx where k is a constant. Show that this function is a solution to the classical expression ((d^2 ψ(x))/ dx^2)= -((2pi)/lambda)^2 ψ(x) What is ...
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196 views

Hydrogen atom in quantum mechanics

I have problems following the calculations in Griffiths' Introduction to Quantum Mechanics (Chapter 4.2.1): If you apply the Schrödinger equation to the Coulomb potential you get the following ...
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3answers
220 views

Is this interpretation of $\psi=\frac{1}{\sqrt{\pi a^{3}}}e^{-r/a}$ correct?

Apologies if this is stating the obvious, but I'm a non-physicist trying to understand Griffiths' discussion of the hydrogen atom in chapter 4 of Introduction to Quantum Mechanics. The wave equation ...
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0answers
225 views

exponential potential $ \exp(|x|) $

For $a$ being positive what are the quantization conditions for an exponential potential? $$ - \frac{d^{2}}{dx^{2}}y(x)+ ae^{|x|}y(x)=E_{n}y(x) $$ with boundary conditions $$ y(0)=0=y(\infty) $$ I ...
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Calculating the error by a small change of the potential in Schrodinger equation

In $\mathbb{R}^3$, consider the time-dependent (non-rel) Schrodinger equation with the potential energy $V(\mathbb{x})$. When a small change(e.g., just a small constant $\delta>0$) of V(x) is ...
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671 views

Schrodinger's equation (explanation to non physicist)

For a report I'm writing on Quantum Computing, I'm interested in understanding a little about this famous equation. I'm an undergraduate student of math, so I can bear some formalism in the ...
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1answer
188 views

Does a particle in a spherically symmetric infinite square well potential exert a force on the inner and outer shell barrier?

For a particle in the potential: $$V(r) = \begin{cases} 0 & \text{a < r < b}\\ \infty & \text{otherwise.} \end{cases}$$ Does this guy in the ground-state exert a force on the shells a ...
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1answer
195 views

Gross-Pitaevskii equation in Bose-Einstein condensates

I was hoping someone might be able to give a approachable explanation of the Gross-Pitaevskii equation. All the sources I've been able to find seem to concentrate on the derivation, and I don't have ...

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