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 compute the expectation value $\langle x^2 \rangle$ in quantum mechanics?

$$\langle x^2 \rangle = \int_{-\infty}^\infty x^2 |\psi(x)|^2 \text d x$$ What is the meaning of $|\psi(x)|^2$? Does that just mean one has to multiply the wave function with itself?
3
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1answer
278 views

What are the properties and characteristics of a single Quantum?

In Quantum mechanics , a quantum of energy called Quanta is origin of everything. In physics, a quantum (plural: quanta) is the minimum amount of any physical entity involved in an interaction. ...
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2answers
2k views

Wave function of Hydrogen Atom [closed]

Wavefunction of a Hydrogen atom is expressed in eigenfunctions as: $$\psi(\boldsymbol r,t=0)=1/\sqrt{14}(2\psi_{100}(\boldsymbol r)-3\psi_{200}(\boldsymbol r)+\psi_{322}(\boldsymbol r) ).$$ Is ...
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4answers
431 views

Are a quantum mechanical system a chaotic (yet deterministic) system?

The title is slightly misleading. I really want to know if the randomness and probabilities observed in quantum mechanics is really just the result of a chaotic (yet deterministic) system. If it is ...
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5answers
1k views

The role of representation theory in QM/QFT?

I need help understanding the role of representation theory in QM/QFT. My understanding of representation theory in this context is as follows: there are physical symmetries of the system we are ...
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3answers
445 views

Takhatajan's mathematical formulation of quantum mechanics

So I began skimming L. Takhatajan's Quantum Mechanics For Mathematicians, and saw the mathematical formulation of QM that he uses (page 51). (The PDF file is available here.) I've only taken a basic ...
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3answers
948 views

If superposition is possible in QM, why do we often assume systems are already in their eigenstates?

My understanding is that an arbitrary quantum-mechanical wavefunction can be written as a linear combination of eigenfunctions of some Hermitian operator, most commonly the Hamiltonian; when a ...
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1answer
638 views

Show that purity = 1 in a pure state

How can you show that for any pure state, the purity = 1? Pure state: $\rho^2 = \rho$ and $Tr(\rho^2)=1$ Mixed state: $\rho^2 = \rho$ and $Tr(\rho^2)<1$
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4answers
3k views

What does superposition mean in quantum mechanics?

What does superposition mean in quantum mechanics? When I say $A+B=C$ (forces). I can mean push something with force $A$ + force $B$ together, and that is same as I push it with force $C$. But when ...
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1answer
2k views

Schrödinger function: Separable wave function with even potential function of x

I have done the Problem 2.1 in Griffiths' quantum mechanics, and it seems not making sense to me. What if the wave function isn't symmetric at all? Then obviously the proof doesn't work. The ...
6
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1answer
676 views

Aharonov-Bohm Effect and Integer Quantum Hall Effect

What is the relationship between Aharonov-Bohm effect and Integer Quantum Hall effect?
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1answer
126 views

Practical meaning of making a measurement/observation in QM?

When an argument like 'measure the spin along the $x$ axis', 'observe the position of a particle' and so on is made, what is the implied experimental procedure? Since laboratory equipment is ...
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3answers
769 views

How can we create superposition in QM?

How can we force a particle (let's say that we know this particle has spin up) to be in a superposition of spin up and down? Wouldn't literally any interaction of it with anything cause it to be in ...
2
votes
2answers
477 views

Does every measurement correspond to an eigenstate of an observable?

In the postulates of quantum mechanics, physical observables are described by Hermitian matrices on the state space of a system. In another of my questions, the measurements of Rydberg-Ritz spectral ...
3
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3answers
298 views

What determines which observables are QM?

Spin, position, and velocity are observables which are QM for quantum particles. My question is, what determines whether an observable is QM or not? For example, why is electric charge not QM? That ...
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2answers
393 views

Classical Communication in Quantum Teleportation

How we could define the classical communication in the quantum teleportation protocol? I mean, classical communication means to send a classical signal. But what happens if we are in an unclear ...
2
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2answers
275 views

How do you assign an observable to spectral lines in Heisenberg's resolution of Rydberg-Ritz?

Ron's comment essentially answers the question below: Here's what I really want to know: Suppose I have an experiment that yields 6 spectral lines corresponding to (one-way) transitions between 4 ...
4
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2answers
568 views

Has BCS Cooper pair condensate been observed in experiment?

Feshbach resonance in s-wave scattering states a BCS Cooper pair condensation at B-field just above the resonance where the scattering length a <0. Just wondering if the condensation has been ...
2
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4answers
619 views

If wave packets spread, why don't objects disappear?

If you have an electron moving in empty space, it will be represented by a wave packet. But packets can spread over time, that is, their width increases, with it's uncertainty in position increasing. ...
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2answers
1k views

Is it possible to recover the old Bohr-Sommerfeld model from the QM description of the atom by turning off some parameters?

Is it possible to recover the old Bohr-Sommerfeld model from the QM description of the atom by turning off some parameters? Can we use Ehrenfest's theorem (or some other scheme) to reduce the QM ...
3
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0answers
310 views

What is the relationship between consistent histories and path integrals?

As can for example be learned from chapter I.2 of Anthony Zee's Quantum field theory in a nutshell, path integrals can be used to to calculate the amplitude for a system to transition from one state ...
7
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4answers
702 views

Does uncertainty imply noncommutativity?

We already know that non-commutativity of observables leads to uncertainty in quantum mechanics cf. e.g. this and this Phys.SE post. What about the opposite: Does uncertainty imply noncommutativity? ...
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2answers
1k views

The definition of entropy in quantum mechanics

I have seen entropy with several different definitions. Like Von Neumann entropy and Rényi entropy, etc. So I am curious why there are so many different definitions in quantum mechanics while only ...
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3answers
2k views

Expectation values of $(x,y,z)$ in the $|n\ell m\rangle$ state of hydrogen?

Expectation values of $(x,y,z)$ in the $| n\ell m\rangle$ state of hydrogen? Does anyone know of a quick way of finding this (if there is even one)? Can I somehow use the relation that: $$\langle r\...
2
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2answers
274 views

Where is noncommutativity in the state-effect formalism of quantum mechanics?

In quantum information theory, one can adopt the basic formalism where every system is given by an operator algebra, state preparation procedures correspond to linear functionals on that algebra (...
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1answer
979 views

Derivation of Bloch's theorem

I'm having a problem following a derivation of Bloch's theorem, looking at a one dimensional lattice with $N$ nodes and spacing a, we impose periodic boundary conditions, meaning that the wave-...
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2answers
187 views
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2answers
319 views

Energy spectrum of a Dirac electron

How do you explain easily "The spectrum of an electron in a repulsive potential " and hence "bound state of charge conjugation" in Dirac hole theory ?
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0answers
183 views

Wilson lines, boundary conditions, surface defects of TQFTs

I asked the following question in mathematics stack exchange but I'd like to have answers from physicists too; I have been studying (extended) topological quantum field theories (in short TQFTs) from ...
2
votes
5answers
456 views

Wave/particle duality

Apologies if this has been asked before (I did check and I believe it wasn't). I have a question about the particle/wave duality of photons (or other particles). Depending on what and how we measure ...
2
votes
1answer
1k views

Solving time dependent Schrodinger equation in matrix form

If we have a Hilbert space of $\mathbb{C}^3$ so that a wave function is a 3-component column vector $$\psi_t=(\psi_1(t),\psi_2(t),\psi_3(t))$$ With Hamiltonian $H$ given by $$H=\hbar\omega \begin{...
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1answer
2k views

Rigged Hilbert space and QM

Are there any comprehensive texts that discuss QM using the notion of rigged Hilbert spaces? It would be nice if there were a text that went through the standard QM examples using this structure.
2
votes
1answer
182 views

Relation between electric charge and gauge parameter of the moduli space of monopoles

I am studying about the moduli space of a 2 monopole system from Harvey's notes, and Manton's paper. In both of these, (Harvey section 6.2), after constructing the Lagrangian for a two dyons system, ...
2
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1answer
13k views

Bound States in a Double Delta Function Potential [closed]

Let $V(x) = −u \delta(x) - v \delta(x − a)$ where $u, v > 0$ correspond to a potential with two $\delta$ wells. Let $v > u$. If $a$ is very large, there is certainly a bound state: the particle ...
0
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1answer
786 views

Charge conjugation in Dirac equation

I need to know the mathematical argument that how the relation is true $(C^{-1})^T\gamma ^ \mu C^T = - \gamma ^{\mu T} $ . Where $C$ is defined by $U=C \gamma^0$ ; $U$= non singular matrix , $T$= ...
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1answer
836 views

Commutation relation with Hamiltonian

How do we get $[\beta , L] = 0$ , where $L$= orbital angular momentum and $\beta$= matrix from Dirac equation?
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1answer
345 views

Why are the equal probabilities for Bell state measurement outcomes essential for “quantum teleportation”?

I've recently been introduced to the basics of finite-dimensional quantum mechanics from a purely mathematical point of view (with a quantum-information theme to it). When discussing quantum ...
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1answer
1k views

Force exerted on potential wall

A particle bound in an infinite potential wall at $x=0$ will apply a force on the wall. For a plane wave and imagining it as a fluid bouncing off the reflection wall at $x=0$, find the force in terms ...
1
vote
1answer
369 views

Symmetry and overlapping of ground states

In a quantum mechanics, there is the following formula to derive the zero energy $E_0$ of a perturbed Hamiltonian $$H = H_0 + V$$ knowing the zero energy $W_0$ of the free Hamiltonian $H_0$: $$E_0 = ...
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6answers
3k views

What constitutes an observation/measurement in QM?

Fundamental notions of QM have to do with observation, a major example being The Uncertainty Principle. What is the technical definition of an observation/measurement? If I look at a QM system, it ...
1
vote
1answer
117 views

How to solve the tranmission probability in an evolution of a quantum system

I've just learned the evolution of some quantum system for about a week, and our homework sometimes something like this. I don't quite have any idea of solving this kind of problem. Can you help ...
2
votes
1answer
179 views

Does wavefunction reach its largest peak near(not in) the classical forbidden region?

As we can see in the picture in this website: http://ctz116.ust.hk/xyli2/images/animation/quchem73.html It's strange that the bound state wavefunction always reach its largest peak near the boundary ...
1
vote
2answers
983 views

A Derivation of Ehrenfest's Theorem in a particular case

What are the missing lines in the integration? $$\frac{\text d \langle {p} \rangle}{ \text{d} t} $$ $$= \frac{\text d}{\text d t} \int\limits_{-\infty}^{\infty} \Psi^* \left( \frac{\hbar}{i}\frac{\...
4
votes
1answer
939 views

Feynman diagrams and Hartree-Fock

I am puzzled by some lines I read in Mattuck's book on Feynman diagrams in many-body problems ( http://www.amazon.com/Feynman-Diagrams-Many-Body-Problem-Physics/dp/0486670473 ) Page 21 (1.14) for ...
6
votes
4answers
1k views

Uncertainty Principle for Information?

I'm not familiar (yet) on how Information theory can be emerged/used in QM/QFT but I was thinking about this question: While we have Heisenberg uncertainty principle on measuring coupled observables, ...
2
votes
3answers
326 views

Zero Point Fluctuations

The total energy of a mode in a quantum mechanical resonator is given by $E_n ~=~ (n+ 1/2)hf$ where $n$ is the number of modes. So when there are no modes or vibrations, i.e. $n=0$, the energy is ...
7
votes
1answer
297 views

Is it possible to make statements about bosonic/fermionic systems by taking the limit $\theta\to \pi$ or $\theta\to 0$, of an anyonic system?

One might naïvely write the (anti-)commutation relations for bosonic/fermionic ladder operators as limits $$ \delta_{k,\ell} = \bigl[ \hat{b}_{k}, \hat{b}_{\ell}^\dagger \bigr] = \...
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votes
1answer
572 views

Basis transformation between eigenstates of harmonic oscillators with different frequency

Given two harmonic oscillators with frequencies $\Omega$ and $\Omega'$, the eigenstates themselves are exactly known. Let's call them $\Psi_n$ and $\Psi'_n$. Is there a compact expression for the ...
3
votes
1answer
455 views

How do eigenstates of harmonic oscillators with different frequencies compare?

Suppose I have a harmonic oscillator with frequency $\Omega_1$ and another one with frequency $\Omega_2$. Is there a simple relationship between the eigenstates of the two? Especially, how would the ...
9
votes
4answers
5k views

Difficulties with bra-ket notation

I have started to study quantum mechanics. I know linear algebra,functional analysis, calculus, and so on, but at this moment I have a problem in Dirac bra-ket formalism. Namely, I have problem with "...