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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Calculating an integral containing a commutator for the hamiltonian and spin for a one electron atom

Problem: Calculate the following integral for a 1-electron atom $ < 2 p_x\alpha \,\, |\,\, [\hat{H},\hat{S_z}] \,\,|\,\, 2 p_x\alpha > $ This is my attempt at a solution: $$ \, ...
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1answer
63 views

Translation Operators

Show that if the wave function $\langle x|\psi\rangle$ is modified by a position-dependent phase $\langle x|\psi\rangle \to e^{\frac{ip_ox}{\hbar}}\langle x|\psi\rangle$ then $\langle x\rangle ...
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1answer
91 views

What's the significance of the difference between the quantum numbers, $\ell$ and $m_{\ell}$?

I know that $m_{\ell}$ is associated with the projection of the angular momentum vector onto the $z$ axis and $\ell$ is associated with the length of the angular momentum vector. To me this implies ...
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1answer
68 views

Quantum Entanglement

My book is generally being quite unclear about something. So firstly I know that if the system is not entangled, we can write its state as $|ab\rangle=|a\rangle|b\rangle$ (if we understand the ...
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27 views

Unitary base transformation applied to continous bases

For discrete vector spaces one can define a unitary base transformation between two complete orthogonal bases $\{ | b_k \rangle \}$ and $ \{ | a_k \rangle \}$ as $$U = \sum \limits_k | b_k \rangle ...
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76 views

Proper way to quantize the string in the light-cone gauge

In many books like Polchinski and Green-Schwarz-Witten the light cone quantization is carried out in a fast way. They just use the virasoro constraint in the light-cone gauge to get the ligh-cone ...
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52 views

Commuting with time evolution operator implies commuting with Hamiltonian

Consider a quantum system (finite dimensional) has overall Hamiltonian: $H_t = H_0 + w(t)H_c$ with $H_0, H_c$ constant in time and traceless and $w(t)$ a, not too badly behaved, function of time. ...
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37 views

Translation Operator

Let $|\psi\rangle \to |\psi'\rangle = \hat{T}(\delta x)|\psi\rangle$ for infinitesimal $\delta x.$ Show that $\langle x \rangle' = \langle x \rangle + \delta x$ and $\langle p_x \rangle' = \langle ...
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70 views

Considering spin angular momentum, what is the magnetic moment of a hydrogen 1s electron, and its energy levels?

This question, posed in a problem sheet that I have been asked to do, has stumped me. I really don't know what to do here. Any help would be greatly appreciated. I know that the magnetic moment of an ...
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44 views

Which phenomenon makes the electrons revolve around the nucleus instead of crashing into it? [duplicate]

If the system has to gain stability, the electrons must crash into the nucleus but that's not the case.
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1answer
35 views

Ignoring spin, what is its orbital magnetic moment of an electron in a hydrogen atom in the 2p orbital?

I know that a magnetic dipole moment is given by $$\mu=\frac e{2m}I$$ and that the angular momentum is $$\frac {m_jh}{2\pi}.$$ However, I have also seen that angular momentum $I$ is given by $$I=\frac ...
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66 views

Why is orbital angular momentum quantized according to $I=\frac h{2 \pi} \sqrt{l(l+1)}$?

I simply have no idea how this result is found $$I=\frac h{2 \pi} \sqrt{l(l+1)}.$$ The result seems to just be dumped in textbooks rather than explained. I can get the result that ...
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1answer
41 views

Amount of entanglement in terms of greatest eigen value for hermitian matrices?

I was reading the paper No Universal Qubit Flipper. In this the paper they show inability to create a universal flipping machine. The method they follow is they take an entangled state between Alice ...
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1answer
59 views

If I were to keep increasing the mass of particles I throw at a double slit, at what point would it stop creating an interference pattern? [duplicate]

Why don't things like tennis balls create interference patterns when thrown at double slits? Where's the limit where it stops interfering?
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1answer
62 views

Meaning of “vacuum state”?

I just learned about $|0\rangle$ and siblings $|0_\gamma\rangle$ and $|0_\infty\rangle$ while studying coherent and squeezed states in a QM class, and I have a question about the meaning of ...
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2answers
81 views

Could Quantum Entanglement be related with gravitational interactions? [closed]

Particles are entangled when they share the same state and their properties become correlated. Interaction with one of them, by say measurement, is experienced instantly by the other. If we measure ...
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1answer
311 views

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?
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2answers
84 views

How do I prove that the del squared operator commutes with the angular momentum operator? [closed]

I need to prove in Cartesian coordinates that $[\nabla^{2},\hat{L_{z}}]= 0$ I know that the angular momentum operator is defined as: $\hat{L_{z}}=x\hat{p_{y}}-y\hat{p_{x}}$ And the del squared is ...
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1answer
66 views

Anticommutator difference [closed]

What is the value of this difference of anticommutators $$\{x^2,p^2\}-(\{x,p\}^2)/2$$ if the commutator $$[x,p]=i\hbar ~?$$ I have tried and obtained a value $$-3\hbar^2/2 - 2i\hbar px.$$ But ...
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33 views

Principle behind Atom Interferometry?

In Laser Interferometry, you propagate a laser beam, split it into two different paths, reflect once, combine it back and deduce the phase difference accumulated from the path difference, from the ...
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30 views

What physical effects cause materialization of a system of particles for a short time?

It is well-known from physics that a photon with enough energy creates a pair of particles: one electron and one positron. This pair of particles can only exist for a short time. This process is ...
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34 views

commutation relations for operators in projected subspaces

I am looking for a consistent re-definition of commutators for certain operators when I work in a projected subspace. Basically, I have a spin defined in terms of 4 Majorana operators $b_{x}$, ...
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2answers
57 views

Schrodinger field and klein gordon field

In the usual Fourier expansion of schrodinger fields \begin{align} \Psi(\vec{x}) = \frac{1}{(2\pi)^{\frac{3}{2}}} \int d^3 k \hat{a}_k e^{-i (wt-\vec{k}\cdot \vec{x})}, \quad \Psi^{*}(\vec{x}) = ...
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19 views

Energy-level choice for electron at photon emission [duplicate]

An electron absorb a photon and jumps to the last energy level.Now when it goes done,How does it choose which level it will step down to?.Also I've come to now that it can omit one step(e.g from 5th ...
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78 views

Prove that $e^{\frac{i\lambda}{\hbar} S_x}S_ze^{-\frac{i\lambda}{\hbar}S_x}=S_z\cos(\lambda)+S_y\sin(\lambda)$ [closed]

Prove using Hadamard's lemma that $$e^{\frac{i\lambda}{\hbar} S_x}S_ze^{-\frac{i\lambda}{\hbar}S_x}=S_z\cos(\lambda)+S_y\sin(\lambda) $$ where $\lambda$ is a complex number. I get: ...
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2answers
61 views

How is CNOT operation realized physically?

I think I understood very well how operations on one qubit are done - if qubit is electron, we just apply magnetic field in direction we want to make spin precess (unitary operations on single qubit). ...
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40 views

Do interaction free experiments violate Quantum Physics?

Although I know that interaction free experiments come under Quantum Physics, Don't the kind of violate the Heisenberg uncertainty principle? Because you get a value without interacting with the ...
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2answers
100 views

Is teleportation possible?

When I just learned about quantum entanglement, I read that if two electrons are in quantum entanglement and placed in two different places, the photon which interacted with one electron is teleported ...
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1answer
88 views

Is de Broglie matter wave a mass or a particle hypothesis?

I'm having difficulty understanding de Broglie matter wave hypothesis. It is a mass or a particle hypothesis? According to de Broglie a particle with mass $m$ moving at a constant speed has an ...
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1answer
89 views

Prove that $(e^{i\lambda A})^\dagger=e^{-i\lambda A^\dagger}$ [closed]

Prove $$(e^{i\lambda A})^\dagger=e^{-i\lambda A^\dagger}$$ where $A$ is an operator. Can anyone explain how to go about this question? Writing it as a power series gets confusing. So basically I ...
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1answer
28 views

Quantum harmonic ocillator and the mean energy U(T)

The energy of a quantum harmonic oscillator is given by: $$E(n)=\hbar\omega\left(n+\frac{1}{2}\right)$$ The canonical partition function is given by: $$Z(T)=\sum_{n=o}^\infty e^{-\beta ...
2
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1answer
59 views

Can All 4-D column matrices be given as tensor product of 2-D column matrices?

I am familiar with entanglement concept. But it feels bit weird to me that all possibilities of a system in a $4$-dimensional vector space cannot be given as tensor product of two $2$-dimensional ...
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2answers
79 views

What is the correct quantization of Hamiltonians of the form $H=x·p$ whose naive quantization is not Hermitian?

As we all learn in Basic Quantum Mechanics (the first quantization), we promote the classical variables to operators. Say, the classical hamiltonian $\frac{p^2}{2m}+V(x)$ to ...
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33 views

A question on intermediate step in deriving gravitational anomaly by Fujikawa's method

In Fujikawa's 'Path integrals and Quantum Anomalies', Eq.(10.26) in the derivation of gravitational anomaly in Chapter 10.1 is puzzling for me. ...
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1answer
59 views

Difference between quantum numbers j and m and the four others

I am confused about what is the difference between the quantum numbers $j$ and $m$ and the other four quantum numbers: the principal quantum number $n$, angular momentum $m$, etc.? From Quantum ...
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1answer
40 views

Charge density of an atomic orbital

Chemistry textbooks on atomic orbitals typically start off with the concept of electrons (viewed as negatively charged point-particles) moving around the nucleus, attracted and bound by the Coulomb ...
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1answer
24 views

Photoelectric effect and work function relation

Let's consider the graph above for two metals $M_1$ and $M_2$ showing relation between photocurrent and potential. First question that I want to ask here is that how do you relate stopping ...
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47 views

How to calculate the expectation value of position/momentum using path integrals?

We have the formula: \begin{equation} \langle F \rangle = \frac{\int Dx \times F[\phi] exp\{i/\hbar S[\phi]\}}{\int Dx \times exp\{i/\hbar S[\phi]\}} \end{equation} Now, I am wondering how a change ...
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1answer
61 views

Covariance of the Dirac Equation

i want to show that the following equation holds: $$ \frac{1}{8}\left[\gamma^{\mu},\omega_{\mu \nu} [\gamma^{\mu},\gamma^{\nu} ] \right] = \omega^{\mu}_{~~~\nu}\,~ \gamma^{\nu} $$ $\gamma^{\mu}$ ...
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1answer
47 views

How to calculate the expectation value of position vector?

$$\psi (\vec{x})=Ae^{-(1/4a^2)|\vec{x}-\vec{x}_0|^2}e^{i\vec{p}_0\cdot \vec{x}/\hbar}$$ The wave function is like this, then how is the expectation value of position vector (not position) calculated? ...
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13 views

plane wave state used for scattering calculation

When we do scattering calculation off a potential barrier, normally we would assume some incident plane wave and then make use of the boundary conditions. But how is this corresponding to the ...
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1answer
88 views

Quantum anharmonic ocscillator $E_0(\lambda)$ curve or table

I am looking for the exact data on $E_0(\lambda)$ for the anharmonicity $\lambda x^4$. The perturbative expansion is the following: $E_0(\lambda)\approx 0.5(1+1.5\lambda -5.25\lambda ...
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2answers
50 views

One dimensional scattering cross section

My quantum mechanics book (http://www-thphys.physics.ox.ac.uk/people/JamesBinney/qb.pdf, pages 94-95 of the pdf, or 86-87 of the book) considers the one dimensional step potential ...
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66 views

How does light oscillate?

Why do we say that electromagnetic wave is oscillating? Or does light propagate really in a wavy form like this image? What is making the photons oscillate and how is it oscillating is it ...
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23 views

What are the reactions that take place inside battery at the quantum level?

I was just studying about how a battery works on the internet and found out that there are reactions of chemicals which make the electrons move. But what exactly happens inside a battery (lets take a ...
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49 views

Is it possible to make an electromagnetic shield?

Is it possible to make an electromagnetic shield to protect something (or ourselves) against things like bullets, human attacks, metals thrown on us or simple dust particles to protect a spaceship ...
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68 views

How can we prove this system is in a stationary state?

I'm having trouble using the given hint to solve the problem. The problem statement is as follows: At instant $t=0$, the probability distribution of a particle under a potential $V(x)$ is ...
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1answer
37 views

How to solve infinite square well with exponential solution (of oscillatory type)?

Given a potential well of $V = 0$ on the interval $(0,L)$ and $V = \infty$ outside the well, I am working to solve the Time Independent Schrodinger Equation $$\dfrac{d^2}{dx^2} \psi= ...
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1answer
50 views

Can an Atom be negatively as well as positively charged simultaneously?

For example there are three atoms- atom A, atom B, atom C. Atom A has 3 electrons, atom B has 4 electrons and atom C has 2 electrons. If we bring together atom A and atom B, in this case atom A is ...
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29 views

Accuracy of quantum mechanical algorithms

Is it possible for a quantum mechanical algorithm to ensure when the wave function collapses that the right answer will be obtained 100% of the time? For this, I am excluding processes such as ...