Tagged Questions
2
votes
1answer
44 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. ...
0
votes
1answer
64 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 ...
1
vote
2answers
46 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
$$\psi(x,0) = ...
0
votes
2answers
57 views
Electron in an infinite potential well
Does this problem have any sense?
Suppose an electron in an infinite well of length $0.5nm$. The state of the system is the superposition of the ground state and the first excited state. Find the ...
0
votes
0answers
55 views
The gauge-invariance of the probability current
It is simple to show that under the gauge transformation $$\begin{cases}\vec A\to\vec A+\nabla\chi\\
\phi\to\phi-\frac{\partial \chi}{\partial t}\\
\psi\to \psi ...
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 $t=0$?
...
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 ...
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 ...
0
votes
1answer
99 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}
...
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
101 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:
$$
...
1
vote
1answer
146 views
Finite, square, potential well
Lets say we have a finite square well symetric around $y$ axis (picture below).
I know how and why general solutions to the second order ODE (stationary Schrödinger equation) are as follows for ...
4
votes
2answers
343 views
Quantum Mechanics: Show that the expectation value of angular momentum does not change with time
The potential is given by $V\left(\left\|(x,y,z)\right\|\right)$, so $[\hat{L}, \hat{H}] = 0$.
Using the definition of $\langle \hat{L} \rangle$ and the time-dependent Schrödinger equation, show that ...
1
vote
1answer
96 views
Constant-dependent potential in radial Schrodinger equation
Studying quantum mechanics, I've found an exercise I don't know how to solve it. Given the radial Schrödinger equation,
$$\left [ \frac{d^2}{dr^2}+k^2-\frac{2m}{\hbar^2}\lambda U\left ( r \right ) ...
1
vote
2answers
469 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 ...
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 ...
2
votes
1answer
308 views
Solving time dependent Schrodinger equation in matrix form
If we have a Hilbert space of $\mathbb{C}^3$ so that a wave function is a 3-component column vector
$$\psi_t=(\psi_1(t),\psi_2(t),\psi_3(t))$$
With Hamiltonian $H$ given by
$$H=\hbar\omega
...
0
votes
0answers
168 views
Force of a particles on a Potential Barrier [closed]
A particle confined by a potential wall exerts some pressure on it. More specifically, suppose that the particle moves in this potential:
$$V(x) ~=~\left\{ \begin{array}{lcc}\text{finite ...
2
votes
1answer
1k views
Bound States in a Double Delta Function Potential [closed]
Let $V(x) = −u \delta(x) - v \delta(x − a)$ where $u, v > 0$ correspond to a potential with two $\delta$ wells. Let $v > u$. If $a$ is very large, there is certainly a bound state: the particle ...
0
votes
0answers
183 views
Finding transcendental equation for the energy of a particle in delta potential well near infinite potential barrier [closed]
I'm having trouble finding the transcendental equation for a particle in a delta potential settled near an infinite potential wall. The potential is given by
$$
V(x) =
\begin{cases}
\infty & x ...
1
vote
2answers
495 views
Barrier in an infinite double well
I am stuck on a QM homework problem. The setup is this:
(To be clear, the potential in the left and rightmost regions is $0$ while the potential in the center region is $V_0$, and the wavefunction ...
2
votes
1answer
362 views
Schrödinger equation with complex potential
In 1 dimension what is the solution of the Schrödinger equation with potential
$$ V(x) = V_r + i V_i $$
Potentials are constant.
1
vote
1answer
307 views
Electron Incident On A Finite Potential Barrier
This is problem 2.8.3 from Miller's Quantum Mechanics For Scientists And Engineers. I'm getting stuck when I try to figure out the wave equation on the right-hand side of the barrier. The original ...
0
votes
1answer
352 views
Bound states for sech-squared potential
I'm working on an introductory qm project, hope somebody has the time to help me (despite the length of this post), it will be highly appreciated.
My goal is to determine the bound states and their ...
2
votes
1answer
521 views
Calculating Ground State Energy in 1D Potential
Given potential $V(x) = Asec(x)$ for $x > 0$. I want to calculate the ground-state energy $E_0$ via the Schrödinger equation.
I'm completely stuck on this one. I've set up the time-independent ...
2
votes
1answer
809 views
How to solve Schrodinger Equation - Tunnelling
I have to solve analitically the Schrodinger equation in one-dimension with a barrier of potential (tunnel effect):
$$ih \frac{d}{dt} U(x,t) = \left[ \left(-h^2 \frac{d^2}{dx^2} \right) + q V(x) ...
1
vote
1answer
240 views
Solving Schrödinger's equation for a specific potential
I am trying to solve this differential equation:
$$-\chi''(\epsilon)+\Big[\epsilon^2+\frac{2F}{hw}\sqrt{\frac{h}{hw}}\epsilon \Big]\chi(\epsilon)=\mu\chi(\epsilon) \tag1$$
This was found ...
-1
votes
2answers
141 views
Show that the energy levels of a particle in a specific potential are $E_n=(n+\frac{1}{2})h\omega-\frac{1}{2}\frac{F^2}{m\omega^2}$ [closed]
A particle of mass m moves on the x-axis under the influence of the potential
$$V(x)=\frac{1}{2}m\omega^2x^2+Fx$$
Can anyone help me, using Schrödinger's equation in one dimension that the energy ...
1
vote
3answers
402 views
One dimensional Schrödinger equation equation with initial condition, finding the probability of the particle's future position
A particle of mass $m$ moves freely in the interval $[0,a]$ on the $x$ axis. Initially the wave function is:
$$f(x)=\frac{1}{\sqrt{3}}\operatorname{sin}\Big( \frac{\pi x}{a} ...
2
votes
1answer
767 views
How to calculate time evolution of a wave function in an 1D infinite square well potential?
A particle in an infinite square well has an initial wavefunction
$$\psi (x,0) ~=~ Ax(a-x) \qquad \mathrm{for}\qquad 0\leq x\leq a.$$
Now the question is to calculate $\psi (x,t)$.
I have ...
2
votes
1answer
264 views
Superposition of wavefunctions
Suppose you have 2 normalized wavefunctions $\psi_1=Ne^{iax}e^{if(x)}e^{i\omega t}$ and $\psi_2=Ne^{-iax}e^{if(x)}e^{i\omega t}$ defined on $x\in [-x_0,x_0]$ and vanishes for $|x|>x_0$. What then ...
2
votes
1answer
1k views
Degeneracies of the first excited state
I have a box with $x,y,z$ all ranging from 0 to $l$. It has $V(x)$=0 inside and =$\infty$ outside. By extending the 1D Schrodinger equation, I have that the allowed energy eigenvalues are ...
3
votes
1answer
194 views
Projection of states after measurement
Continuing from the my previous 2-state system problem, I am told that the observable corresponding to the linear operator $\hat{L}$ is measured and we get the +1 state. Then it asks for the ...
0
votes
1answer
263 views
Two-state system problem
Given a 2-state system with (complete set) orthonormal eigenstates $u_1, u_2$ with eigenvalues $E_1, E_2$ respectively, where $E_2>E_1$, and there exists a linear operator $\hat{L}$ with ...
2
votes
1answer
2k views
How to solve this Schrödinger equation?
I am taking an intro level quantum mechanics class. Our textbook gives a problem like this:
The deuteron is a nucleus of "heavy hydrogen" consisting of one proton and one neutron. As a simple ...
