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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58 views
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 ...
2
votes
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. ...
6
votes
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 ...
-1
votes
0answers
29 views
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 ...
0
votes
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 ...
3
votes
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 ...
1
vote
2answers
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
$$\psi(x,0) = ...
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votes
2answers
57 views
Electron in an infinite potential well
Does this problem have any sense?
Suppose an electron in an infinite well of length $0.5nm$. The state of the system is the superposition of the ground state and the first excited state. Find the ...
0
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0answers
55 views
The gauge-invariance of the probability current
It is simple to show that under the gauge transformation $$\begin{cases}\vec A\to\vec A+\nabla\chi\\
\phi\to\phi-\frac{\partial \chi}{\partial t}\\
\psi\to \psi ...
4
votes
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 ...
0
votes
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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votes
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 ...
3
votes
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 ...
5
votes
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?
2
votes
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 ...
3
votes
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 ...
4
votes
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 ...
4
votes
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
$$
\frac{\mathrm{d}}{\mathrm{d}x}\left[ (1-x^2) ...
1
vote
1answer
86 views
The Klein–Gordon equation
As we know that the Schrödinger equation presents basis of Quantum Mechanics and analogy with Newton second law in Classical Mechanics, I thought that relativistic interpretation of Schrödinger ...
0
votes
1answer
98 views
Potential step and its transmission / reflection
Lets say we have a potential step with regions 1 with zero potential $W_p\!=\!0$ (this is a free particle) and region 2 with potential $W_p$. Wave functions in this case are:
\begin{align}
...
2
votes
2answers
97 views
Why the hydrogen radial wave function is real?
Why the hydrogen radial wave function is real?
Is it a coincidence?
3
votes
1answer
133 views
How does a state in quantum mechanics evolve?
I have a question about the time evolution of a state in quantum mechanics. The time-dependent Schrodinger equation is given as
$$
i\hbar\frac{d}{dt}|\psi(t)\rangle = H|\psi(t)\rangle
$$
I am ...
0
votes
0answers
43 views
Wave equations for two intervals at Potential step
Lets say we have a potential step as in the picture:
In the region I there is a free particle with a wavefunction $\psi_I$ while in the region II the wave function will be $\psi_{II}$.
Let me ...
0
votes
1answer
100 views
Energies and numbers of bound states in finite potential well
Hello I understand how to approach finite potential well (I learned a lot in my other topic here). However i am disturbed by equation which describes number of states $N$ for a finite potential well (
...
2
votes
2answers
285 views
Plotting $\psi$ for finite square well potential
Lets say we have a finite square potential well like below:
This well has a $\psi$ which we can combine with $\psi_I$, $\psi_{II}$ and $\psi_{III}$. I have been playing around and got expressions ...
1
vote
1answer
109 views
Finite potential well - transcendent equation for even solutions
I have a finite square well like the one on the picture below:
I have done some calculations on it and got a transcendental equation for even solutions which is like this:
$$
...
1
vote
0answers
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 $ ...
0
votes
1answer
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 ...
1
vote
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 ...
0
votes
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 ...
0
votes
0answers
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 ...
-1
votes
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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votes
2answers
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 ...
1
vote
1answer
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 ...
3
votes
2answers
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:
...
1
vote
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?
3
votes
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 ...
2
votes
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 ...
1
vote
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,
$$\left [ \frac{d^2}{dr^2}+k^2-\frac{2m}{\hbar^2}\lambda U\left ( r \right ) ...
1
vote
1answer
179 views
Solving the 1-D Schrodinger equation for a free particle: Confused about 2 possible general solutions
I am following Griffiths' Introduction to Quantum Mechanics, as well as an online lecture that follows a different book, and both sources give different equations for the general solution of the 1-D ...
0
votes
1answer
385 views
Solving the time independent Schrodinger equation: Does a complex solution make sense?
In my notes, I have the Time Independent Schrodinger equation for a free particle
$$\frac{\partial^2 \psi}{\partial x^2}+\frac{p^2}{\hbar^2}\psi=0\tag1$$
The solution to this is given, in my notes, ...
1
vote
2answers
467 views
Particle in infinite potential well which is doubled in size at $t_0$
I am currently studying for an exam in Quantum Mechanics and came across a solution to a problem I have trouble with understanding.
The Problem:
A Particle sits in an infinite potential well ...
0
votes
0answers
49 views
Proving One-dimensional Wave with Classic Expression [closed]
A general form of a one-dimensional wave is ψ(x) = sin kx where k is a constant. Show that
this function is a solution to the classical expression
((d^2 ψ(x))/ dx^2)= -((2pi)/lambda)^2 ψ(x)
What is ...
1
vote
2answers
196 views
Hydrogen atom in quantum mechanics
I have problems following the calculations in Griffiths' Introduction to Quantum Mechanics (Chapter 4.2.1):
If you apply the Schrödinger equation to the Coulomb potential you get the following ...
2
votes
3answers
220 views
Is this interpretation of $\psi=\frac{1}{\sqrt{\pi a^{3}}}e^{-r/a}$ correct?
Apologies if this is stating the obvious, but I'm a non-physicist trying to understand Griffiths' discussion of the hydrogen atom in chapter 4 of Introduction to Quantum Mechanics. The wave equation ...
6
votes
0answers
225 views
exponential potential $ \exp(|x|) $
For $a$ being positive what are the quantization conditions for an exponential potential?
$$ - \frac{d^{2}}{dx^{2}}y(x)+ ae^{|x|}y(x)=E_{n}y(x) $$
with boundary conditions $$ y(0)=0=y(\infty) $$
I ...
1
vote
0answers
53 views
Calculating the error by a small change of the potential in Schrodinger equation
In $\mathbb{R}^3$, consider the time-dependent (non-rel) Schrodinger equation with the potential energy $V(\mathbb{x})$. When a small change(e.g., just a small constant $\delta>0$) of V(x) is ...
9
votes
3answers
671 views
Schrodinger's equation (explanation to non physicist)
For a report I'm writing on Quantum Computing, I'm interested in understanding a little about this famous equation. I'm an undergraduate student of math, so I can bear some formalism in the ...
1
vote
1answer
188 views
Does a particle in a spherically symmetric infinite square well potential exert a force on the inner and outer shell barrier?
For a particle in the potential:
$$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 ...
4
votes
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 ...
