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 ...

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111 views

Can the expectation value of the square of momentum be negative?

I've been solving a problem in quantum mechanics, and I was deriving the standard deviation of $P$, knowing that $\langle P\rangle=0$. Because $\Delta P=\sqrt{\langle P^2 \rangle - \langle P \rangle ...
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0answers
28 views

Why can create a particle with $k$ by using $c_{−k,−σ}$? [duplicate]

I have asked the same question before but without answer. In Mudelung's book, Introduction to Solid-State Theory, I have a confusion about the statement. enter image description here He showed the ...
0
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1answer
60 views

Addition of Angular Momentum

I am tring to find the eigenvectors of a two spin system, with $j_1=3/2$ and $j_2=1/2$. To start, $$m_1 =-3/2,-1/2,1/2,3/2$$ $$m_2=-1/2,1/2$$ For $j_1$, there are 4 possible states, and 2 possible ...
2
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1answer
67 views

In the Dirac equation, do $\alpha$ and $p$ commute?

The Dirac Hamiltonian is given as $H = \vec \alpha·\vec pc + \beta mc^2$ , Do the alpha and beta operators commute with the momentum operator? If yes then how?
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5answers
139 views

How does quantum world affect us and why should I care about it? [duplicate]

We live in a world which is much larger than quantum world. The laws of quantum physics are not valid. While I am pressing the keys on my laptop, I have 100% certainty that I am writing what I really ...
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1answer
138 views

Lie group of Schrodinger Wave equation

In Ballentine's book on quantum mechanics (in 3rd chapter), he introduces the symmetry transformation of Galilean group associated with Schrodinger equation. Now the Galilean group as such has 10 ...
2
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2answers
66 views

How to deal with mean field method in antiferromagnetism?

There are lots of ways to apply the mean field method to deal with the Ising model whose ground state is a ferromagnetic state. Hence, it is easy to find the order parameter named magnetization to ...
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0answers
61 views

Variational principle

In the LMTO method, the interstitial region is approximated by plane waves and the muffin tin region of the potential by solutions to the radial Schrodinger equation. In using the variational method ...
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2answers
168 views

Why $-i\hbar\vec\nabla$ for momentum in quantum mechanics, while $m\vec{v}$ in classical mechanics?

I am a little bit confused when thinking of the momentum representation in QM and CM. In QM, momentum is represented as $-i\hbar\vec\nabla$, while in classical, momentum is represented as $m\vec{v}$. ...
4
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1answer
122 views

Trace of an operator matrix (Quantum computation and quantum information)

I'm reading the book Quantum computation and quantum information by Mike & Ike and I'm stuck at 2.60/2.61. There, the author says that, given the operator $A|ψ⟩⟨ψ|$, its trace is: $${\rm ...
2
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1answer
70 views

Why is the camera not the culprit? [duplicate]

Perhaps I am completely wrong, but as I understand it our observation of a system can affect the outcome. The example I remember is the double slit experiment where electrons behave as a wave at ...
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2answers
71 views

Momentum Operator in Quantum Mechanics

1) What is the difference between these two momentum operators: $\frac{\hbar}{i}\frac{\partial}{\partial x}$ and $-i\hbar\frac{\partial}{\partial x}$? How are these two operators the same? My ...
3
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1answer
95 views

Understanding the Selection Rules of a Spin-Forbidden, Magnetic Dipole Transition in Molecular Oxygen

I am studying the transition from the second excited electronic state of molecular oxygen, $b^1\Sigma_g^+$ , to the ground state, $X^3\Sigma_g^-$. I know that the ground state has total angular ...
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1answer
56 views

Comparing two infinite sets

All the linearly independent eigenfunctions of the parity operator $\mathcal{P}$ form an infinite set and all the linearly independent eigenfunctions of the unit operator $\bf 1$ also form an ...
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0answers
16 views

Quantum computing records (storage times)

Long storage times for qubits will be integral in the construction of a scalable quantum computer. This leads me to ask the current state of affairs in our ability to store qubits. Namely, what is the ...
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3answers
91 views

Quantum entanglement: does it necessarily imply superluminal information transfer? [duplicate]

From what I understand, information is communicated instantly between two quantum-entangled particles regardless of the spatial distance between them. However, does this necessarily imply superluminal ...
5
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4answers
92 views

Why can interference from two independent sources be observed?

Having read this question and answers to it, I've learned that somehow two light beams from independent sources can actually produce interference pattern, if the properties of their sources are good ...
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7answers
843 views

Why can't we have a wave of particles?

I understand the nature of light can be complex and has extensive theories/experimental data. We hear light can be both a wave and particle, so why can't it be both, a wave of particles?
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2answers
66 views

Matrix elements of linear operators - orthonormal basis required?

In an early linear algebra class of mine, I learnt that a linear map $\mathcal{A}$ acting on a vector space could be represented by a matrix $A_{ij}$ according to the rule: $$\mathcal{A}({e_j}) = ...
6
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2answers
83 views

Precise meaning of composition of ket and bra, e.g. $|\psi\rangle\langle\psi|$

I'm currently studying density matrices, and have been frequently coming across the construction $$|\psi\rangle\langle\psi| \,.$$ What is the formal meaning of this composition? I understand ...
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0answers
22 views

For the two identical particles scattering, How can i identify two particles are bosons or fermions?

If two particles are scattered. How can i know those two particles are bosons or fermions?
5
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1answer
95 views

Why don't we use Hamilton-Jacobi method in QM?

In classical mechanics, we usually try to find a set of coordinates by Hamilton-Jacobi method to transform the Hamiltonian to zero such that the coordinates are conservations. However, we never try ...
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3answers
43 views

Difference between expectation value and probability amplitude?

I was given a wave equation. I know that probability amplitude is the eigenvalue of an observable operating in a state. $$H| \psi\rangle = h| \psi\rangle$$ where $h$ is the probability amplitude of ...
6
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1answer
101 views

What are the restrictions on the Hamiltonian in QM?

In quantum mechanics, we usually write the Hamiltonian as: $$\hat{H}=\hat{T}+\hat{V}$$ But in classical mechanics, there are several reasons why it would not have this form: We've chosen some ...
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0answers
69 views

Difference between expectation values of $L^2$, $L_z$ and measuring $L^2$, $L_z$

I was given with this hydrogen radial wavefunction $$ R_{21} =\left(\sqrt{\frac{1}{3}}Y^0_1 + \sqrt{\frac{2}{3}}Y^1_1\right) $$ and was asked to find a) What are the expectation values of the ...
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1answer
69 views

Expectation value of energy from the position state of hydrogen atom [closed]

I was given with the position state of hydrogen atom: $$ R_{21} =\left(\sqrt{\frac{1}{3}}Y^0_1 + \sqrt{\frac{2}{3}}Y^1_1\right) $$ I am getting confused about getting the expectation value of ...
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2answers
122 views

Size of a photon

When detecting radio waves in space, we use very large telescopes or arrays of telescopes. But according to QM, aren't photons point particles when measured? Does a photon with a large wavelength ...
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0answers
23 views

Spherical Harmonic projection on axis

I am trying to solve for the Spherical harmonics $Y^m_{l=1}$ with a second axis at an angle $\alpha$ with respect to the z axis. Then this can be used to find the probability that a particle with ...
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0answers
42 views

How does $\bar{r}\times(\bar{\nabla}\times) - \bar{\nabla}\times(\bar{r}\times)$ relate to the orbital angular momentum operator?

When I attempted to calculate the following by hand $$\bar{r}\times(\bar{\nabla}\times\bar{F}) - \bar{\nabla}\times(\bar{r}\times\bar{F}),$$ I noticed some of the terms I extracted looked similar to ...
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0answers
33 views

Including special relativistic effects in momentum in Heisenberg's Uncertainty Principle

I've been told that an electron is somewhere within the space of $10^{-10}m$ and am supposed to find the uncertainty in its velocity. Simply applying $m\Delta x \Delta v \geq \frac{h}{4\pi}$ results ...
3
votes
1answer
149 views

Harmonic Oscillator potential, proof that Gaussians remain Gaussians?

I read in several papers that for a Harmonic Oscillator Hamiltonian in the time dependent Schrödinger equation a Gaussian wave packet remains Gaussian. Unfortunately I could not find any proof for ...
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3answers
132 views

On Group Theory: Symmetry Groups and Our Interest

Over the past few years, I've been doing a lot of self education in the Quantum Mechanics and General Relativity, and of course, there are mathematical elements of both doctrines that are matrices. ...
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1answer
123 views

Energy conservation limited by uncertainty principle

The way I learned it from practicing Fourier analysis and signal processing besides quantum mechanics, is that Energy conservation cannot be achieved in short time scales, and that limits energy ...
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5answers
91 views

EPR paradox: instantaneous vs very fast?

An EPR quantum experiment can be explained by instantaneous collapse of the wave function regardless of the distance separating a pair of entangled particles. But do we have the certainty that the ...
0
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1answer
68 views

Aufbau principle in modern quantum theory

What is the rigorous definition of the Aufbau principle and the mathematical model used for its description? From Wikipedia, we have that the principle postulates a hypothetical process in which an ...
12
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5answers
323 views

What does periodicity of $e^{-iHt/\hbar}$ mean in physical terms?

The unitary time evolution operator $U(t)=e^{-iHt/\hbar}$ has some distinct flavour of periodicity to it because of $e^{x+2\pi i}=e^x$. Is this periodicity reflected in any way in physical systems? ...
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1answer
58 views

Significance of magnetic translation operator defined in fractional QHE's description

What is the significance of the magnetic translation operator used in describing the Fractional Quantum hall effect? I was following Anthony Leggett's lecture video in which he defines these operators ...
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0answers
41 views

Why cannot we apply perturbation theory in Born-Oppenheimer approximation

In Weinberg's Lectures on Quantum Mechanics, he mentions Unfortunately, we cannot simply use first-order perturbation theory, with $T_{nuc}$ taken as the perturbation and the state vectors ...
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40 views

Quantum Excitations

In the context of quantum field theory, is the schrodinger or dirac equation actually describing some sort of an actual wave in some field like light in EM field ? So all particles are actually waves ...
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1answer
82 views

What is $\langle \sigma_\mu \rangle$ $\langle \sigma_\mu \rangle$ for the Pauli Matrices?

What is \begin{align} \sum_{\mu=0}^{3} \langle \sigma_{\mu} \rangle^2 = ? \end{align} $\sigma_{\mu}$ are the Pauli matrices. The Bra-Ket notation is used in this question: \begin{align} \langle ...
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1answer
58 views

Many worlds interpretation

I think many worlds interpretation is inconsistent with the EPR paradox. Quantum mechanics says that particles are really in more places at the same time and the particle is really only probability ...
0
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0answers
16 views

Clarification about “projections and correlations” in Heller's Quantum Gravity

The footnote (bottom) for the following paragraph from “Creative Tension: Essays On Science & Religion” by Michael Heller, has “such a projection, being always “onto,” switches off all possible ...
2
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1answer
95 views

Are they the same thing: Wigner distribution in quantum Boltzmann equation and Wigner function in quantum optics?

We know that quantum Boltzmann equation (QBE) is an equation of motion for the interacting Green's function $G^<(\vec{x}_1,t_1;\vec{x}_2,t_2)\equiv\mathrm{i}\langle ...
2
votes
1answer
48 views

What are “parity considerations” in deciding the form of the Hamiltonian?

In "introductory Quantum Optics", by Gerry and Knight, the Jeynes model is considered. In this model of electron-EM field interaction the electron is approximated by a two state system ($\lvert ...
0
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0answers
31 views

motion of electrons [duplicate]

Do electrons move randomly, with no preference of directions? And why electrons don't fall into the nucleus? About this question, I read the article on Chemistry wiki, which says that when electron ...
8
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1answer
183 views

Double double-slit experiment

Suppose I have a double double slit experiment. That is, I have an electron gun in the center, that shoots entangled pairs of electrons in opposite directions, one to each double slit. I tried to ...
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1answer
79 views

Correspondence between wave function and state vector

I am confused with connection between state $| \psi \rangle$ of a quantum system and corresponding wave function $\psi(x)$ (at a given time). I have been told that for every state $| \psi \rangle$ we ...
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0answers
51 views

Does quantum mechanics require classical measurement apparatus?

I am trying to learn quantum mechanics and I have a question. Landau, in his quantum mechanics book says that it is in principle impossible to formulate basic concepts of quantum mechanics without ...
0
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1answer
63 views

Confusion about state of a quantum system

I am confused with the concept of state of a quantum system. First postulate of QM ussualy says that the wave function of the system contains all information about the state of the system. But reading ...
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0answers
66 views

Time Evolution Operator in Interaction Picture (Harmonic Oscillator with Time Dependent Perturbation)

1. The problem statement, all variables and given/known data Consider a time-dependent harmonic oscillator with Hamiltonian $$\hat{H}(t)=\hat{H}_0+\hat{V}(t)$$ $$\hat{H}_0=\hbar \omega \left( ...