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Tagged with semiclassical wavefunction
34 questions
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Action as the complex phase of the wave function of quantum mechanics
I heard that the action of classical mechanics can be seen as the complex phase of the wave function of quantum mechanics
$$\psi=\rho \exp\left(\frac{iS}{\hbar}\right)\tag1$$
I am more familiar with ...
- 904
4
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2
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129
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Semi-classical Quantum Ping-Pong in an infinite well potential
The general one particle state in a simple infinite well of size $L$ is a superposition of all the Hamiltonian eigen-states:
$$\tag{1}
\psi(x, t) = \sqrt{\frac{2}{L}} \sum_{n = 1}^{\infty} c_n \, e^{-\...
- 7,687
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1
answer
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Relation between the wavelength and the particle-wave duality
I will go straight into an example. Let's take the case of an electron of mass $m$ confined in an infinite 1D box of width $a$. Solving the Schrödinger equation and pay attention to the boundary ...
- 123
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2
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128
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Motion/momentum of a wave packet
I'm reading Dirac's "Principles of quantum mechanics" right now, being a little confused about the following part: (Chapter V$\S$31: "The Motion of Wave Packets").
He's making the ...
- 255
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1
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Intution for the physical meaning of high energy limit of a quantum states and uniform distribution in phase spacehow of a particle
Zeev Rudnick state in his talk Quantum Ergodicity for the Uninitiated (around 12 minute 40 second mark at the last text section of the slide) that a "a possible interpretation of the statement ...
- 125
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The WKB approximation and boundary conditions
To estimate a quantization rule using the WKB approximation, one is usually working in the 'classically allowed region'. You apply the WKB approx in the middle of the region and use the Airy ...
- 472
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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 ...
- 31
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WKB approximation derivation for $E<V$
I understand that we can write any complex wavefunction on polar form $A\exp(iθ)$ with both $A,θ$ real. Following the logic of Griffiths on WKB (here, page 291):
We write the energy wavefunction in ...
- 379
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3
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Contradiction in my understanding of wavefunction in finite potential well
Most things like to occupy regions of lower potential. So the probability amplitude should be higher in a region of lower potential. I denote the potential by V.
However, we also know that the kinetic ...
- 553
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How to derive Landau level with semiclassical approach?
I'm trying to derive the Landau level by applying semiclassical dynamics and the time-dependent Schrodinger equation. From that, I success to derive $E = \hbar\omega_c n$, but I fail to derive the ...
- 31
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What is the meaning/justification for the matching conditions in the WKB approximation?
Consider the time-independent Schrodinger equation in one dimension with a potential $V(x)$ at fixed energy $E$. In the WKB approximation, we obtain solutions in the classically allowed region (i.e. ...
- 131
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1
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Expansion of $\phi (x)$ in the derivation of WKB approximation
In the derivation, we assume the eigenfunctions of $H$ to have the form $$\psi (x)=e^{i \phi(x)/\hbar}$$ where $\phi (x)$ is allowed to take any complex value.
But then suddenly we assume this ...
- 6,666
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5
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Would a high energy Hydrogen atom start emanating electromagnetic radiation?
We know that the total energy of the hydrogen atom is proportional to the inverse of the square of the principal quantum number $n$:
$$E_n \propto -\frac{1}{n^2}$$
So at high quantum numbers the ...
- 35.5k
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1
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Assumption made for the WKB approximation in radial coordinates [duplicate]
I was thinking the other day, if you had the Schrodinger equation in 3-dimensions, and had a spherically symmetrical potential. Ie.:
$$-\frac{ℏ^{2}}{2m}∇^{2}ψ+V(r)ψ=Eψ$$
Then you could simplify the ...
- 307
1
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1
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WKB application on symmetric potential well
I am a little confused how one can find a wave function by using WKB approximation? I do know the oscillation frequency $$\Omega ~=~ {2E\over h}{\rm Re} \langle L|R \rangle~=~ {E\over \pi\hbar}{\rm Re}...
user288166
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1
answer
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Wavepacket for hydrogen atom?
We normally observe classical behaviour due to the time dependent schrodinger equation in simple quantum systems when we introduce 'Gaussian wavepackets' which have bell shaped uncertainty in energy, ...
user86425
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1
answer
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Problems deriving the Quantum Hamilton-Jacobi equation
This is my first question at Physics SE so please be kind. I am well versed in the etiquette over at Math SE, but not so much here. Anyway, I thought this question was better suited to this site ...
- 291
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1
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When is a logarithm of the wavefunction well-defined?
It is sometimes convenient to write the wavefunction as
$$
\Psi(x,t)~=~ e^{\Phi(x,t)}
$$
and then work with $\Phi$ instead. This is particularly sensible in the context of the WKB approximation, where ...
- 1,141
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1
answer
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WKB approximation difficulty - deciding what term to neglect
Consider the following quantum well:
Region 1 is a classically forbidden region, and hence the WKB wave-function will take the form of equation
$$\psi(x) = \frac{C}{\sqrt{q(x)}}e^{+\int_b^a q(x')dx'/\...
- 988
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1
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Wavefunction of a particle on a ring ($E > V$) using WKB method
For a particle on a ring (with radius $R$ and changing angle $\theta$) with only kinetic energy ($V=0$) we get the expressions for the wavefunction (normalized) and eigenvalues $$\Psi_n (\theta) = \...
- 43
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2
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WKB approximation for a particle on a ring $(E>V)$
So I have this problem where a particle is on a ring of perimeter L, and the coordinate of the particle is denoted by $s$, $0<s<L$. There is a nonzero potential which varies with $s, V(s)$, and ...
- 163
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WKB for $E > V(x)$
When we use the WKB method, at least when I learned it, all of our examples had $V(x) > E$ at some point, allowing for turning points.
Say we have some $V(x) < E$ for all $x$. How would we ...
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2
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The link between discrete energy level in quantum mechanics and harmonic series in Acoustics
Consider a quantum square potential well with infinite depth:
$$
V(x)=\begin{cases}
0, &|x|<a \\
+\infty, &\text{otherwise}.
\end{cases}$$
Solving the Schodinger equation of a particle with ...
- 1,365
1
vote
1
answer
373
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Connection formulas: Why do we assume asymptotic behavior of the Airy functions?
The derivation is obtained from Introduction to Quantum Mechanics by Griffiths
Let assume the turning point occurs at $x=0$, then the WKB solutions right and left to the turning point are:
$$
\psi=\...
- 277
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1
answer
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What is the wavelength at the classical turning points using WKB Approximation? [closed]
According to what I know is that a classical turning point in Newtonian Mechanics is a point where a particle has a zero kinetic energy (Total energy is equal to potential energy) and must be ...
1
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0
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127
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WKB near turning point by means of complex integration (Landau & Lifshitz, Quantum Mechanics) [duplicate]
The question is basically about section 47. in Landau's Quantum Mechanics (non-relativistic theory)
everything is fine until the sentence (at the beginning of the second page) where he says: it is ...
- 357
1
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2
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128
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$\hbar \approx 0$ and the spread of QM wave function
Is there a direct mathematical method to show that if a quantum wave funtion is initially sharply localized, then it will stay sharply localized if $\hbar \approx 0$? In that case the Ehrenfest ...
- 733
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WKB Connection Formula Clarification
From Griffiths, if we have some potential $V(x)$ and energy $E$ such that $E=V(0)$ where $V(x)<E$ for all $x<0$ and $V(x)>E$ for all $x>0$. In the patching region, Griffiths uses only one ...
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Landau & Lifshitz's Approach (contour method) on the WKB connection formulas
Background of the question (see pp. 161, section 47 in Landau & Lifshitz's quantum mechanics textbook Vol3, 2nd Ed. Pergamon Press).
We have the following potential well
$$U(x)\leq E \quad\text{...
- 111
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2
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Why does it mean to say a quantum state's "penetration" is comparable to the distance over which it fluctuates?
Consider the following diagram of an energy eigenstate in a harmonic potential.
The textbook from which I got this image says
Also, in the classical limit the quantum mechanical probability ...
1
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1
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How do you get quantum Hamilton-Jacobi equation from Schrödinger equation? [closed]
I am reading "The undivided universe: an ontological interpretation of quantum theory" and cannot understand this derivation.
From the Schrödinger equation:
$$
i\hbar \frac{\partial\psi}{\partial t} =...
- 11
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1
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What is a 'turning point' in WKB and why does it fail at that point?
What is meant by a classical turning point in quantum mechanics and why does the WKB approximation fail at that point?
- 1,375
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1
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Bohr-Sommerfeld quantization for different potentials
Let's have Bohr-Sommerfeld quantization for one-dimensional case:
$$
\int \limits_{a}^{b} p(x)dx ~=~ \pi \hbar (n + \nu ).
$$
Here $p(x) = \sqrt{2m(E - U)}$, $a, b$ are turning points, and the area ...
user8817
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1
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Bohr-Sommerfeld quantization condition from the WKB approximation
How can one prove the Bohr-Sommerfeld quantization condition
$$ \oint p~dq ~=~2\pi n \hbar $$
from the WKB ansatz solution $$\Psi(x)~=~e^{iS(x)/ \hbar}$$ for the Schroedinger equation?
With $S$ the ...
- 4,797