Quantum mechanics describes the microscopic properties of nature in a regime where classical mechanics no longer applies. It explains phenomena such as the wave-particle duality, quantization of energy and the uncertainty principle and is generally used in single body systems. Use the quantum-field-...

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177 views

Gauge transformation of vector potential multiplies wavefunction by phase

Consider an electron in an electromagnetic field with scalar and vector potentials $\phi, \mathbf{A}$. Suppose for simplicity that $\mathbf{A}$ is time independent. Suppose also that we know the ...
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0answers
52 views

Efficient Method for Multiplying Angular Momentum Operators

I'm doing a calculation that involves canonical symmetrization of angular momentum. For example: $H_{\text{classical}} = J_x J_y \rightarrow \hat{H}_{\text{quantum}} = \frac{1}{2}(\hat{J_x}\cdot \...
2
votes
2answers
159 views

Quantum Operators: An Identity

I came across the following neat property: For an operator $\hat{A}$ which is a linear combination of creation and annihilation operators, we have: $$ \langle e^{\hat{A}} \rangle = e^{\langle \...
12
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2answers
311 views

Why does the conjugated $\pi$ bond not violate the Pauli Exclusion Principle?

Let's look at the molecule 1,3 butadiene: $CH_2=CH-CH=CH_2$ and number the carbon atoms 1 to 4 from left to right. The bonds between 1 and 2 and between 3 and 4 are double bonds: each ...
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0answers
41 views

How to calculate the partial trace [duplicate]

Can anyone help me in explaining how this example below get the reduced density matrix from the density matrix in bipartite system. $$\rho =\frac{1}{4}\begin{pmatrix} 1 & 1 & cos(\frac{\alpha}...
3
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2answers
369 views

Double slit experiment; evidence of wavefunction collapse

This video shows the change of a photon's interference pattern in real time of the Young's single and double slit experiment. In this video it is claimed that by adding a detector to view how the ...
5
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2answers
185 views

Spin drift velocity?

I am currently reading this Phys Rev paper by H C Torrey. In this paper, he derives the Bloch equations with an additional diffusion term. He says that the current density is given by $$\mathbf j_{\...
0
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1answer
45 views

Why Does there Have to be Linearity in Ket and Skew Symmetry?

I'm reading Shankar's "Principles of Quantum Mechanics," and on page 8 he states that one axiom in Dirac notation is linearity in ket, and because they are also skew symmetric there is anti-linearity ...
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1answer
42 views

Expectation energy for a quantum harmonic oscillator

At 59:14 in this video, the expectation value of the energy of a harmonic oscillator is $$ \langle E \rangle = \int ||\tilde{\Psi}(p)||^2 \frac{p^2}{2m}\ \mathrm dp + \int ||\Psi(x)||^2\frac{m\omega^2}...
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1answer
55 views

Reduced Density operator in matrix form

I already read book of Quantum Computation and Quantum Information by Nielsen and Chuang according to reduced density operator and I already understand how to do the reduced density using Dirac ...
4
votes
1answer
304 views

Integral over a product of two Green's functions

Need some help here on a frequently encountered integral in Green's function formalism. Forgive me since I am a junior student. I have an integral/summation as a product of a retarded and advanced ...
3
votes
1answer
60 views

What does partial (non-maximum) quantum entanglement mean?

When quantum systems are entangled, they have a "grade of entanglement" which can be quantified e.g. as the entropy of entanglement. There also are states of "maximum entanglement", e.g. the Bell ...
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2answers
200 views

why does the frequency of a wave remain constant?

They say the frequency of a wave is its fundamental character, thus remain constant throughout its propagation regardless the medium through which it travels. Could anyone explain why frequency of ...
0
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3answers
112 views

What is the cause of quantum entanglement? [duplicate]

I understand the idea of quantum entanglement - where what happens to one particle in one location instantly effects another particle in another location, even if separated by millions of miles. But ...
2
votes
1answer
146 views

Strange implication of the relativistic invariance of the Dirac equation

At least as normally formulated, the law of transformation of a wave function solution of the Dirac equation to another inertial frame seems to indicate that if observer 1 is certain the particle is ...
3
votes
1answer
120 views

Replacing fermionic operators with their Fourier transform and boundary conditions

In the section 4.1 of Quantum Computation by Adiabatic Evolution, Farhi et al proposes a quantum adiabatic algorithm to solve the $2$-SAT problem on a ring. To compute the complexity of the algorithm ...
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2answers
76 views

The counter-intuitive time scales in atomic physics and nuclear physics

Compare atomic physics and nuclear physics. The interaction in the latter is much stronger than that in the former. However, the typical spontaneous emission time scale in atomic physics is on the ...
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3answers
685 views

Entanglement, Bohr-Einstein Debate, Bell's Inequality

On BBC episode The Secrets of Quantum Physics (Part 1) Jim Al-Khalili explains quantum mechanics for the layman. In the first half, he does a very good job; in the second half, either he thought his ...
4
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2answers
78 views

Is there a relation between spin and the spin group?

In Quantum Mechanics spin appears as one type of angular momentum. Indeed, in Quantum Mechanics one angular momentum on the state space $\mathcal{E}$ is a triplet of observables $\mathbf{J}=(J_1,J_2,...
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1answer
33 views

How does a hydrogen ion gas cool?

Ok I understand that a hydrogen gas of non-ions at a temperature higher than its surroundings exists with many excited electrons. These electrons, either spontaneously or due to collisions, will ...
0
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1answer
42 views

Why does Carbon-12 have zero nuclear spin?

While studying NMR theory, my textbook explained that only nuclei with odd mass numbers are NMR active because they have non-integer spin quantum numbers and nuclei with an even mass number and atomic ...
29
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8answers
5k views

Is (rest) mass quantized?

I learned today in class that photons and light are quantized. I also remember that electric charge is quantized as well. I was thinking about these implications, and I was wondering if (rest) mass ...
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1answer
214 views

Explain confinement energy for a particle in a box

What is a "particle in a box"? How does confinement energy equal kinetic energy? How does confinement energy relate to the spectrum energy (the absortion/emission between energy)?
3
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1answer
37 views

Reflection and refraction of single photon incident on a glass slab

If a single photon strikes a glass slab of certain thickness, can we make prediction whether it would reflect or refract? On which factor the reflection or refraction of single photon through such a ...
1
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1answer
45 views

Angular momentum in annihilation $n\overline{n} \rightarrow \pi^0 \pi^0$

Consider the annihilation of a neutron by an anti-neutron $$ n\overline{n} \rightarrow \pi^0 \pi^0 $$ so that the initial relative angular momentum is zero. Because the spin of neutrons is $1/2$, $J_i$...
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2answers
98 views

Why does a electric Potential have to be real, but not a Potential in quantum mechanics?

So I had this Problem when I had to learn about classical electromagnetism: Why is it, that we use complex numbers when calculating stuff, but in the end only the real part is important (for example ...
2
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1answer
233 views

What is many-body bound state?

Bound state by definition is a state when particles are bounded together, so then "many-body bound state" would be bound state for a system of many bodies. Then I have several puzzles: is the state ...
3
votes
1answer
62 views

Tensor product representation of $SO(3)$ in the Hilbert space of particle with spin $S$

For a particle with a spin $S$, the rotation operator is given by $$ e^{iJ_i\theta/\hbar} $$ where $J_i$ is the component of the total angular momentum along the direction of the rotation axis. The ...
4
votes
3answers
2k views

When Eigenfunctions/Wavefunctions are real?

When the Hamiltonian is Hermitian(i,e. beyond the effective mass approximation), generally under which conditions the eigenfunctions/wavefunctions are real? What happens in 1D case like the finite ...
3
votes
5answers
189 views

What happens when two wavefunctions meet?

Apologies for the over-broad question(s), but I'm having a hard time finding out where to look to answer these myself: If a particle is a wavefunction describing a probability amplitude distributed ...
0
votes
1answer
50 views

Can a quantum mechanical system have more than one wave-function?

I was told that a quantum mechanical system is completely determined by its wave function. But superposition principle says that given two wave functions of some system, a linear combination of them ...
3
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1answer
38 views

Protocol for solving time independent Schrodinger equation

Just a short question about the protocol for solving the time-independent Schrodinger equation for different potentials and the reasons for accepting and rejecting solutions. Take for example the ...
1
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0answers
45 views

How does the filling of the 2p orbitals occur?

When electrons enter the 2p orbitals, electrons of the same spin occupy the 2p orbitals first and then electrons of the opposite spin fill up the orbitals. Why is that? My professor told me that there ...
5
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3answers
135 views

Can a physical wavefunction be non-smooth (its first derivative is discontinuous)?

Here's an argument that might support the statement that such a non-smooth wavefunction is not physical: You cannot add a finite number of smooth functions to get a non-smooth function. By fourier ...
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3answers
971 views

Long time deviations from exponential decay in radioactivity

Are there any examples of common substances whose decay is not exponential? We're used to thinking about radioactivity in terms of half-lives. This is a concept that makes sense only for a decay that ...
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0answers
24 views

WKB approximation to find energy levels of a step potential

Suppose the following potential: $$ V(x) = \begin{cases}V_0 & 0<x<\frac{a}{2} \\ 0 & \frac{a}{2}<x<a \\ \infty & \text{otherwise} \end{cases} $$ Also, assume that for every ...
4
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6answers
24k views

In the famous Einstein's Photoelectric effect, why does the intensity of light not raise the kinetic energy of the emitted electrons?

The work function of any metal is no doubt constant for it is related to electromagnetic attraction between electrons and protons. However on increasing the intensity of any light source the kinetic ...
2
votes
1answer
102 views

Can a particle pass through a point where wave function is zero?

Let's consider an infinite square well. In the first exited state there is a node at the middle of the well (i.e. wave function and thus probability of finding the particle is zero there). If I ...
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2answers
272 views

Double slit experiment

I can't understand the relation between Schrödinger's Cat and this amazing experiment of the double slit. It seems like in the double slit we know if we observe the particle the wave function ...
0
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0answers
37 views

“Instantaneous” time stepping with time dependent Hamiltonian Schrodinger equation

The Schrodinger equation for time-dependent Hamiltonian is $$i\hbar\frac{d}{dt}\psi(t) = H(t)\psi(t) \, .$$ I know that the "instantaneous" solution of this equation is $$\psi(t+dt) = e^{-\frac{i}{...
2
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0answers
58 views

Rigorous way of box normalisation

This is follow up from an answer to my previous question about unitarity in rigged Hilbert space. As it turns out, that there is no idea of unitarity in rigged Hilbert space (hence no meaningful QM ...
0
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2answers
30 views

Book Recommendation: Quantum optics

Could you suggest me a list of books for understanding Quantum Optics for students who have studied Introductory Q.M.(such as Griffiths). It would be grateful if you distinguish between readable one(...
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0answers
28 views

Planck's temperature - why is there a maximum? [duplicate]

Why do the laws of physics break after Planck's temperature?
2
votes
1answer
36 views

Relationship Between Magnetic Dipole Moment and Spin Angular Momentum

I am reading Introduction to Quantum Mechanics 1st edition by David J. Griffiths and I have a couple questions about this section on page 160. A spinning charged particle constitutes a magnetic ...
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0answers
32 views

Identical bosons with spin interactions eigenstates

Suppose that we have two particles where each of them has s=1 and it is in a harmonic oscillator potential and there is also a spin interaction. The hamiltonian of the system is :$$H=\frac{p_1^2}{2m}+\...
15
votes
7answers
1k views

Why is a Hermitian operator a “quantum random variable”?

To me, as a stupid mathematician, a random variable is a measurable function from some probability space $(\Omega, \sigma, \mu)$ to $(\Bbb{R}, B(\Bbb{R}))$. This makes sense. You have outcomes, events,...
0
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2answers
104 views

Can quarks have anti-colors? [closed]

What is the reason that the color properties we call red, green and blue have become tied to quarks, while what we call anti-red, anti-green and anti-blue has become tied to anti-quarks? Do note that ...
2
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0answers
30 views

Distribution of quantum beating among spectral frequencies

Let's imagine that we have 3-level system: ground state $\vert 0 \rangle$ and two excited states $\vert 1 \rangle$, $\vert 2 \rangle$ with similar energies $\hbar \omega _1$ and $\hbar \omega _2$ ...
3
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2answers
91 views

How to visualize an electron existing in two different places at the same time?

Let's consider a hypothetical situation where there are two electrons. The first electron is in superposition, simultaneously existing in two different locations. Let the locations be ...
2
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1answer
124 views

Modern interpretation of wave-particle duality

As far as I understand, in the early days of quantum theory there was quite a lot of debate over how to interpret what it meant for a quantum mechanical object to exhibit both wave-like and particle-...