All Questions
Tagged with potential quantum-mechanics
556 questions
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Formal demonstration of bound and continuous spectra in a 1D potential problem
I understand that if a particle is in a confining potential, its spectrum is discrete. When the particle's energy is greater than the potential, the spectrum becomes continuous, allowing the particle ...
0
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1
answer
51
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Why the probability density for a finite potential well is more inside the well compared to outside where potential is zero for higher value of $n$?
Can somebody explain to me why the probability density is higher in this case inside the barrier region when it should be more where the potential is less?! max(V) is 200 I've normalized it so that to ...
4
votes
1
answer
253
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Double and single delta-function potential well energy similarity
I am working through Griffiths QM at present, and I came across a question that asks me to find
"the bound state energies in the limiting cases (i) $a \to 0$ and (ii) $a \to \infty$ (holding $\...
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83
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Is Sakurai's Modern Quantum Mechanics explanation that the oscillation period between $|S\rangle$ and $|A\rangle$ is infinite wrong?
[Errors that persist in the 3rd edition of Sakurai's textbook?]
The content dealing with the symmetry double-well potential contains an error in the coefficient of $|A\rangle$ in $|R,\,t\rangle$, ...
- 11
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0
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31
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$\pi$ phase shift upon reflection in quantum wells
Is there a similar phenomenon to the $\pi$ phase shift experienced by light upon reflection from a medium of lower to higher refracted index for particles in different potentials?
For instance, does a ...
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-1
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1
answer
52
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How does 1D Schrödinger equation arise out of the postulated 3D Schrödinger equation and solving 1D particle using 3D Schrödinger equation?
I've stumbled upon this question when I was trying to solve the Schrödinger equation for a particle confined to a 1D line with some given time independent potential $V(x)$.
The energy eigenstates ...
- 109
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44
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Hydrogen atom energy: $n$ vs $l$ (QM)
I've come across a problem that states the following: let two (separate) particles be subject to a central potential $V(r)$. Their reduced radial function is depicted in the following image:
Which ...
- 1,880
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1
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52
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Reflection of quantum particle colliding with a potential barrier
Let a quantum particle be subject to a one dimensional step potential barrier $V$ such that:
$$V(x)=\begin{cases}0, \ x<0 \\ V_0, \ x>0\end{cases}$$
where the particle's energy is $E>V_0>0$...
- 1,880
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0
answers
67
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Probabilistic reformulation of classical Simple Harmonic Oscillator
As an interesting exercise, I was wondering whether we could reformulate classical mechanics in such a way that we could use the same mathematical paradigm we use in quantum mechanics. I'll expose it ...
- 1,880
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1
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212
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Why is probability outside the infinite square well zero? [duplicate]
In an infinite square well, potential energy is given below, why is the probability of finding a particle in the position of infinite potential energy zero?
$$V(x)=\begin{cases}
0,& \text{if } ...
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0
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56
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Normalizable solutions to the time-independent Schrödinger equation are real [duplicate]
In Griffiths Introduction to Quantum Mechanics (3ed.) problem 2.1, we are asked to prove that the normalizable solutions to the time-independent Schrödinger equation can always be chosen to be real, ...
- 131
4
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2
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228
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Parity of a bound state determined by potential
Some time ago in my QM class, we were working with an infinite well potential, and my professor told us we could know beforehand the bound states we were going to obtain for said potential would have ...
- 1,880
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1
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48
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How do I find the wavefunction equation for a translated potential? [closed]
Let me explain myself. In my case, I know the wavefunction equation of a infinite-U potential which has the form:
$$V(x)=\begin{cases} V_0,\ \ |x|\leq \frac a2\\ \infty, \ \ |x|>\frac a2\end{cases}$...
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1
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74
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Why does the curvature change with respect to the sign of the wave function?
In quantum mechanics, in the context of studying unidimensional potentials, we studied this case of potential
And to analise the possible looks of the wave function $\phi$, we divided in 3 regions, ...
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1
answer
74
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Boundaries of finite potential well
I'm currently studying the simplest models of potentials in QM, and I've run across an apparent inconsistency in my textbook: when describing a finite potential as a piecewise function, I've noticed ...
- 1,880
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102
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Exactly solvable Schrödinger equations on the circle?
Are there any (famous?) periodic potential functions in 1d, $V(x) = V(x+L)$, so that the Schrödinger equation with periodic boundary conditions $\psi(x)= \psi(x+L)$ is exactly solvable?
I can do it if ...
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2
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87
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Particle in a box - standing wave [closed]
Why can't we consider that a particle in a box is an example of a standing wave?
The ends are fixed by the fact that the potential outside is infinite. The only difference being that it is a de ...
- 1,131
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1
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197
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Ansatz for wavefunction infinite square well with linear perturbation $\alpha \cdot x$
Suppose we have an infinite square well extending from $0<x<L$ and a particle in its ground state. However, the infinite square well contains a linear perturbation $\alpha \cdot x$. The ...
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1
answer
72
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Expectation value of momentum in a particular eigenstate
Suppose a particle is in time independent potential and NOT in any superposition of eigenstates. Then expectation value of position is time independent and expectation value of momentum should always ...
- 353
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1
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141
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Solution of $p$-wave time independent Schrödinger equation with a simple negative potential
I'm currently self-studying quantum mechanics and have encountered a challenge regarding higher angular momentum wave functions $\phi(r)$ on whether the corresponding Schrödinger equation has a bound ...
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2
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147
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Rectangular barrier embedded in Infinite well
Consider the problem:
Let a particle be subjected to the following 1-dimensional potential:
$$
V(x) = \begin{cases}
V_0 > 0 & 0 \leq x < b \\
0 & b < x < a \\
...
- 402
1
vote
0
answers
69
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The derivative of the wave function at the boundary of the wall with potential $U_0$ ($m_1 \neq m_2$) [closed]
In the potential barrier, if a particle in an environment with a potential of $U_1$ hits a potential barrier with $U_2$ and suppose its energy is less than $U_2$.
Now my question is that if the mass ...
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1
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357
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For the bound state to exist why must the potential $V(x)$ have at least one minimum which is lower than $V_1$? [closed]
In chapter 4 of Zettili's book Quantum Mechanics Concepts and Applications he shows this figure:
Then states:
For the bound states to exist, the potential $V(x)$ must have at least one minimum which ...
1
vote
1
answer
48
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Wavefunctions of quantum point contacts
I have a question about the potential and wave functions of discrete energy levels of quantum point contacts. Assuming one has a rectengular potential well:
Can the potential of the QPC be assumed to ...
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1
answer
67
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Can the potential $V$ in the TDSE depend on the wavefunction?
I'm working on some derivations for a homework problem and need to verify whether $V$ is an operator dependent on the wavefunction or if $V$ is just some function.
If I have the following [𝛙(r,t)* V𝛙...
- 719
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1
answer
159
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Quantum infinite square well
In the context of an infinite potential well with boundaries at $(-a,a)$, where the potential is defined as follows:
\begin{equation}
V(x) =
\begin{cases}
0, & \text{if } -a \leq x \leq a \\
...
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0
votes
1
answer
128
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Relation between ground state energy and the potential
I am given the following Hamiltonian:
$$H = \frac{p^2}{2m} + \lambda|x|^3$$
where $\lambda$ is a positive constant.
Is there a relation between the ground state energy of $H$ and $\lambda$ i.e. is ...
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votes
2
answers
276
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What is the difference between the potential energy and potential function in quantum mechanics?
In quantum mechanics, we study particles in various systems, such as an infinite potential well, a finite potential well, potential barriers, potential steps, harmonic oscillators, and so on. In all ...
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137
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Applications of infinite square well/particle in a box
I know of only two instances where the infinite square well is an adequate model for experimental behaviour: the absorption wavelengths of cyanine dyes, and extremely small semiconductors to which ...
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1
answer
114
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What happens on a finite negative well when the particle has smaller energie than the potential well? [duplicate]
This is a common exercise in the classroom. They always check the cases when $E>0$ and $V<E<0$ for a finite negative well $V<0$.
But what about the case $E<V$. What happens here?
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4
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1k
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How does a rectangular potential barrier work (in quantum mechanics)?
I'm having a hard time grasping the concept of 'potential barrier'. I'm currently concerned about a rectangular potential barrier (I have attached a picture).
The classical analogy given is of a ball ...
- 1,306
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2
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408
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Why is probability density function constant with time? (1D Potential Well) [closed]
(I asked a very similar question already, but the core idea is very different in both)
Here are the two equations Im concerned with
$$\psi = \sqrt{2\over a}\sin\Big(n\pi {x\over a}\Big)$$
$$E = {n^2\...
- 1,306
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5
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Particle in 1D Infinite potential well = a ball
Here are the two equations I'm concerned with:
$$\Psi = \sqrt{\frac2a}\sin\left(n\frac{\pi x}a\right)$$
$$E = n^{2}\frac{\hbar^2π^2}{2ma^2}.$$
If we have a ball with mass 1 kg, confined in a 1 m ...
- 1,306
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2
answers
112
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An interesting (but standard for sure) potential well
Consider the potential well with $$U(x) = \begin{cases}-\dfrac{\hbar^2}{m}\kappa_{0}\delta(x), & |x|<a\\+\infty, & |x|\geq a\end{cases}$$
I want to find $E_{n}$. First of all, $U(x)$ is ...
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0
answers
95
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Can the WKB approximation be used to get eigenenergies for negative potential 'barriers'?
I recently took a course that discussed the WKB approximation for linear potential. In class and in the exercises we only looked at pretty simple potentials that are just a constant times |x|. What I ...
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1
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For a particle in a 1D box, what is the expectation value of momentum for a particle that has tunnelled through the walls and escaped?
For a particle in a box, where the walls of the box have a finite (i.e. not infinite) potential
energy, what is the expectation value of the momentum of a particle which has tunnelled
through the wall ...
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1
answer
117
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How limiting a potential affects the energy eigenvalues?
I am asking myself, how constraining a potential changes the energy eigenvalues.
With the WKB-Approximation-Method one can derive that the dependence of the eigenenergies regarding a potential $V(x) \...
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1
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163
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Finite quantum harmonic oscillator and existence of a ground state
I am having some problems with a finite, shifted quantum harmonic oscillator potential, and the theorem that states:
Any attractive potential in one dimension must have at least one bound state.
Let'...
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0
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102
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How to solve the Schrodinger Equation on a sphere with a step potential? [closed]
I am currently trying to solve the Schrodinger equation on a sphere that is subjected to an axially symmetric step potential given by:
$$
V(\theta)=\begin{cases}
0~~~~~~~~\text{if }~~ \theta_{min} <...
1
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0
answers
65
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Experimental realization of rectangular potential barrier
I was reading a QM book where they put the following figure aiming to explain how to realize experimentally a potential barrier/well. In the left (barrier) and right (well) plots they show a circuit ...
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2
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276
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What happens step-by-step (and why) when a particle tries to escape an infinite potential well?
I am aware that the following question might be quite elementary. My background is mainly in mathematics and my physics education is limited to high-school level material (discounting analogues made ...
- 125
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1
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143
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Will the probability for tunnelling go completely to zero? [closed]
According to quantum mechanics, the probability for quantum tunnelling (of an object) never become completely zero, no matter how "big" is the height and the thickness of the barrier.
...
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Schrodinger Equation: Time Dependent, Periodic Potential $$V(x,t)=\begin{cases}V_0x&:t\in[0,\frac{T}{2})\\-V_0x&: t \in [\frac{T}{2},T) \end{cases}$$
Imagine that we have a particle in a cylinder of finite length and neglible radius. We can then assume that the system is axisymmetric and can be solved in one dimension.
Let us consider a time ...
- 185
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1
answer
125
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Wavelength of wavefunction graph [closed]
So my professor loves to make like 10 exam questions where we interpret some graphs of quantum wave functions. Now I really do not understand how he reads off the wavelength for instance in these ...
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6
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3
answers
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Boundedness of a Hamiltonian and when does a Hamiltonian have a spectrum?
In the context of Quantum Field Theory we put restrictions on the potentials we can use. One argument is boundedness. If the potential is unbounded, for example $V(\phi) = \phi^3$, then `the field can ...
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2
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187
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Why is the Bohm quantum potential considered a potential?
In Bohmian mechanics, the term $$\begin{equation}
Q = -\frac{\hbar^2}{2m}\frac{\nabla^2 R}{R}
\end{equation}$$
is regarded as the quantum potential term. However this is merely a term from the real ...
- 3
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1
answer
267
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Variational Method for A Symmetric double well Potential
I am given a set of trial wave functions of the form
$$
Φ_n^{\pm}(x)=Ψ_{n}(x-α)\pm Ψ_{n}(x+a)
$$
Where $Ψ_n$ are the $n$th Harmonic oscillator wavefunctions. in order to approximate the energy levels ...
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0
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30
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Can potentials be used to transmit energy and information?
The famous Aharonov-Bohm effect displays the potential of the physical implications of different potential gauges in em theory.
I saw very few experimental and theoretical investigations into this ...
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1
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46
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Is there anything fundamentally different between these two representations of an infinite square well?
The first representation is what I would say the more typical one:
$$V(x) = \cases{0 & 0<x<a \\ \infty &else} $$
But it could also be:
$$V(x) =\cases{-\infty & 0<x<a \\ 0 &...
- 3,468
2
votes
2
answers
200
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Infinity potential well [closed]
For infinity potential well we take the analogy with nucleus. Inside which proton and neutron are bound in principle, if it is an infinite potential well, so particle should not come outside the ...
