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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How to derive or justify the expressions of momentum operator and energy operator?

It has been noted here$\! { \, }^{\text(1, 2)}$, for instance, that $$\mathbf{F} = \frac{d}{dt}\!\!\biggl[ \, \mathbf{p} \, \biggr]$$ is true in all contexts. Likewise, in notable contexts it is ...
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709 views

Harmonic Oscillator Expectation Value

In Calculating the expectation value of the quantum harmonic oscillator, I've come across a problem for finding $\left \langle x \right \rangle$ for the coherent state $\left| \alpha \right \rangle$ $...
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1answer
187 views

Is there a formalism for talking about diagonality/commutativity of operators with respect to an overcomplete basis?

Consider a density matrix of a free particle in non-relativistic quantum mechanics. Nice, quasi-classical particles will be well-approximated by a wavepacket or a mixture of wavepackets. The ...
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1answer
319 views

Selection rules in Stark effect

The energy level of an electron could be shifted by an electric field. $\langle n, l,m|[L_z,z]|n^{\prime},l^{\prime},m^{\prime}\rangle=(m-m^{\prime})\hbar \langle n, l,m|z|n^{\prime},l^{\prime},m^{\...
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3answers
538 views

Why is ground state $| 0 \rangle$ of harmonic oscillator a coherent state?

Is ground state $| 0 \rangle$ of harmonic oscillator a coherent state just because it minimize the uncertainty product? What is the intuition of this. I don't quite understand the significance of the ...
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65 views

Why does $\langle a_-\alpha|\alpha\rangle = \alpha $for harmonic oscillator

Why does $\langle a_-\alpha|\alpha\rangle = \alpha$ doesn't the ladder operator lower the $\alpha$ so that it became $\sqrt{\alpha}*\delta_{\alpha, \alpha_{-1}}$ Maybe its because that \alpha can be ...
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1answer
107 views

Time-ordered calculation for equal time

I have a question about how to calculate the following expectation value: $$\langle0|\mathcal{T}\{{a^{\dagger}}(0,0) a(0,0)\}|0\rangle$$ where $|0\rangle$ is the ground state and $a^{\dagger}(x,t)$ ...
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2answers
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The harmonic oscillator - ladder operators

Reading from Griffiths. I have got two questions. First, the halmiltonian operator that used to find the energy eigenvalue in only harmonic oscillator is: $$H={\hbar}w(a_-a_+-\frac{1}{2})$$ and $$H={\...
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729 views

Harmonic Oscillator Energy to Momentum Expectation Value

If we are given a wave function written in terms of harmonic oscillator energy eigenfunctions how can we determine the maximum possible momentum expectation value? It's a combination of the first two ...
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Does the Time Evolution Operator Commute with any Other Operators?

Does the time evolution operator in quantum mechanics commute with any other operators, with a commutator of zero? Also, what exactly is the utility of the time evolution operator, is it more ...
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308 views

Why does $[xp_{y},x]$ commute?

I'm looking at a solution in my book that says $[xp_{y},x]$ commutes. Does bracket notation imply: $[A,B]=AB-BA$ so that $[xp_{y},x]=xp_{y}x-xxp_{y}$ Taking the comment from Max Graves and ...
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Entropy increase vs Conservation of information (QM)

Unitarity of quantum mechanics prohibits information destruction. On the other hand, the second law of thermodynamics claims entropy to be increasing. If entropy is to be thought of as a measure of ...
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1answer
158 views

Sign Paradox in Berry's Phase

Suppose we have normalized states $| n(\vec{R})\rangle$ indexed by continuous variable $\vec{R}$. Then fixing our choice of gauge and ignoring dynamic phase, the phase difference between two states ...
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1answer
281 views

What are the Time Operators in Quantum Mechanics? [duplicate]

I don't understand at all what the time operators are in quantum mechanics. I thought that given a wave function, because it's a function of time, we could simple put in any time in the future to find ...
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5answers
184 views

Is $\langle\psi_1|p\psi_1\rangle$ necessarily 0 for eigenstates? [closed]

Is $\langle\psi_1|p\psi_1\rangle$ necessarily 0 for harmonic oscillator eigenstates? If $\Psi(x,t)= c_0\psi_0(x)e^{-iE_0t/\hbar}+c_1\psi_1(x)e^{-iE_1t/\hbar}$, is the following true? Where $p$ is ...
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1answer
149 views

How do I find the expectation value given only the eigen energy? [closed]

Let $|n \rangle$ denote the $n^{th}$ stationary state of the harmonic oscillator, with energy $E_n = \hbar \omega(n+\frac{1}{2})$ How would I find $\langle x\rangle$ and $\sigma_x$ I know that $$\...
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1answer
209 views

Modulus Square of the Gaussian Wave Packet for uncertainty in $p$

Upon evaluating the integral (2.67) and obtaining the complex valued equation given in box 2.4, the author performs the modulus square to obtain the Gaussian distribution (2.68). How does one go about ...
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2answers
148 views

Does measuring the operator of a wave function collapse the wave function to the measured eigenstate?

Suppose you have a state described by the wave function $\psi(x) = \phi_1(x)+2\phi_2(x)+3\phi_3(x)$ , where the $\phi$s are normalised eigenfunctions of a Hermitian operator $\hat{O}$ with eigenvalues ...
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124 views

Wave function interaction

If you have two or more wave functions that represent electrons or other charged particles, how would the force on one be calculated based on the charge of the others.
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869 views

Hilbert Space of (quantum) Gauge theory

Since quantum Gauge theory is a quantum mechanical theory, whether someone could explain how to construct and write down the Hilbert Space of quantum Gauge theory with spin-S. (Are there something ...
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1answer
800 views

Why are orthogonal functions and eigenvalues/functions so important in quantum mechanics?

The mathematics and physics we have studied so far at university are heavily focused around the idea of orthogonal functions, orthogonality, sets of solutions, eigenvalues and eigenfunctions. Why ...
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2answers
632 views

Normalising a wavefunction where $\psi$ is equal to a sum of functions [closed]

The wavefunction $\psi(x)$ = $\phi_1(x)$ + $2\phi_2(x)$ + $3\phi_3(x)$ is to be normalised. The functions $\phi_1(x)$, $\phi_2(x)$, $\phi_3(x)$ are normalised eigenfunctions of a Hermitian operator $\...
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1answer
891 views

Quantum Mechanics Lx operator [closed]

Show that if the state $ \rvert\gamma\rangle $ is real, then the expectation value of each component of the angular momentum is zero. Does this imply the angular momentum is zero? My Work: $$ \...
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1answer
281 views

Matrix operations on Quantum States in a composite quantum system

Intro (you may skip this if you're an expert, I'm including this for completeness): Say I have two bases for two systems, The first is a spin-1/2 system $|+\rangle = \left(\begin{array}{c} 1\\0 \...
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0answers
111 views

How do we show that photons generated by a constant electric current are distributed according to a Poisson distribution?

I saw the answer sometimes ago in a book "Quantum Electronics" or similar title. I don't remember the author since I lost the book. The book sets ( I believe so ) a constant electric current $I$ in a ...
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1answer
643 views

Is it possible to derive Schrodinger equation in this way?

Let's have wave-function $\lvert \psi \rangle$. The full probability is equal to one: $$\langle \Psi\lvert\Psi \rangle = 1.\tag{1}$$ We need to introduce time evolution of $\Psi $; we know it in ...
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1answer
170 views

Do systems with level crossings have unstable eigenbases?

It's folklore dating back to von Neumann and Wigner that time-dependent Hamiltonian systems tend not to have level crossings of their energy eigenvalues. However, we can of course consider smoothly ...
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0answers
351 views

Is total angular momentum conserved in particle interaction?

Imagine that two electrons interact by exchanging a virtual photon. I know that the total energy and linear momentum of the two electrons is conserved by the interaction. Is the total (orbital) ...
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2answers
186 views

Angular momentum representation

It is well know that, using position representation $$\langle r\lvert L\rvert \psi\rangle =r \times (-i\hbar\nabla\langle r|\psi\rangle )=r \times (-i\hbar\nabla\psi(r)).$$ However, I read ...
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1answer
372 views

Going left or going right of the free particle wave function?

In this excerpt from Griffith's quantum mechanics book: THE FREE PARTICLE We turn next to what should have been the simplest case of all: the free particle [$V(x) = 0$ everywhere]. As you'll ...
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159 views

Solution of QM tasks by using asymptotics

When we solve QM tasks by solving the Schrödinger equation, such as tasks about a particle in a Morse potential, a Poschl-Teller potential and many others, we usually find approximations (lets call ...
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A few parity questions for simple harmonic oscillator

I think I understand that the solution to the Schrodinger equation for the SHO is based on the Hermite polynomials (and the Guassian function). The solution set of all even Hermite polynomials are a ...
3
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2answers
392 views

Formulation and probability of a wave-function [closed]

I have got this problem where I have been given the following wave function: $$\Psi = 0\quad\text{if}~|x| > a\quad\text{and}\quad A(a^2-x^2)\quad \text{if} \quad |x|< a$$ Now the first question ...
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3answers
648 views

Is the uncertainty principle valid?

The uncertainty principle says that the product of the uncertainties in position and momentum can be no smaller than a simple fraction of Planck's constant $h$. Several articles lately suggest this ...
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2answers
252 views

Observable Operator on a Superposition?

I'm probably missing something obvious and basic here but I can't make sense of certain usages of Observables as present in basic treatments of Quantum Mechanics that i've come across. $$ \hat{A}|\...
9
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1answer
442 views

Quantum version of the Galton Board

If classical particles fall through a Galton Board they pile up in the limit of large numbers like a normal distribution, see e.g. http://mathworld.wolfram.com/GaltonBoard.html What kind of ...
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0answers
119 views

Usage example of stabilizer codes QEC

This question directly follows the previous one about $X$ stabilizers and phase-flip errors: Practical example of stabilizer codes Let's now consider a second part of the quantum circuit that is ...
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1answer
108 views

Pair production at high laser intensity?

Using high laser intensity to produce electron-positron pair, is it still required interaction with nucleus as is the case when gamma rays are used? What causes the pair creation ?
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223 views

Aren't all electrons the same? So what about electron that absorbs photon?

I learned that electron absorbs a photon and goes into higher energy state. But also all electrons are identical. What is a difference between the electron in low orbital energy state? and the high ...
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2answers
2k views

Quantum Commutator Identities

Question: Prove that $p^2$ and ${\bf r}\cdot {\bf p}$ commute with every component of ${\bf L}$ using the identity $$[{\bf p},{\bf e}\cdot {\bf L}]=i\hbar\, {\...
3
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2answers
4k views

Proving that the hermitian conjugate of the product of two operators is the product of the two hermitian congugate operators in opposite order

I have reach a step in a problem of my quantum mechanics textbook that requires me to prove the following. $$\hat{A}=(\hat{Q}\hat{R})^{\dagger} = \hat{R}^{\dagger}\hat{Q}^{\dagger}$$ I tried to ...
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3answers
106 views

Help with understanding an Operator definition

The operator $\hat{F}$ is defined by $F\psi(x)=\psi(x+a)+\psi(x-a)$ Does this mean $\hat{F}=(x+a)+(x-a)$ and that $\hat{F}$ is operating on $\psi(x)$?
4
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4answers
531 views

Classical Limit in Quantum Mechanics

Suppose I have a wave function $\Psi$ (which is not an eigenfunction) and a time independent Hamiltonian $\hat{\mathcal{H}}$. Now, If I take the classical limit by taking $\hbar \to 0$ what will ...
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1answer
350 views

Heisenberg uncertainty principle - question [closed]

A beam of particles each having mass $m$ and velocity $v$ in the incident on a circular hole of radius $b$ located on a screen. If another screen is placed at a distance $D$ from the hole, determine ...
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1answer
95 views

Electron in strong magnetic field [closed]

What if we apply a very strong magnetic field to an electron so that it's position be a constant. Then if it's position is constant, it's momentum will also be a constant. But it is in violation of ...
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2answers
170 views

Basic Interpretation of Compostion of Observables and their Measurement

Given two (or more) observables $A, B$ which commute one can construct a third observable $C= A \circ B$. If $\psi$ is a common eigenvector of $A, B$ with eigenvalues $\lambda_1, \lambda_2$ then it is ...
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200 views

Particle Spacing in a Vacuum

Four questions: (To start off, I know very little about physics it isn't even funny (I probably use a ton of wrong terms here and leave out vital information, if so I will try to edit it in as you ...
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3answers
232 views

Why are we living in the $q$ part of the phase space?

In Hamilton mechanics and quantum mechanics, $p$ and $q$ are almost symmetric. But in the real world, the $p$ space isn't as intuitive as the $q$ space. For example, We can uniquely identify a person ...
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1answer
237 views

Quantum fluctuations and expanding universe

As far as I understand, Hawking radiation is formed at the edge of a black hole, when a particle/anti particle pair is formed and one of the particles falls into the black hole before the particles ...
4
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2answers
345 views

Can the quantum eraser experiment result indicate a 'computed universe'?

The quantum eraser experiment tells us that a photon shot at two slits is a wave, unless you measure which slit is taken and you do not destroy the measurement result. I've found this very similar to ...