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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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 ...
2
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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
533 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
354 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 ...
-1
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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 ...
5
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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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1answer
201 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 ...
6
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3answers
233 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
242 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
355 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 ...
7
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3answers
3k views

Why Don't the Ladder Operators Commute?

I have two problems with ladder operators. The first is that I feel they should somehow result in measurable things. The asymmetry of applying the plus operator versus the minus operator is very ...
3
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2answers
4k views

How To Use Ladder Operators?

I'm studying for a test in quantum mechanics and I'm having a hard time understanding how to use ladder operators. There are no examples in my text book, only definitions that I can't understand how ...
2
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3answers
167 views

Curie temperature and magnetization?

If an ferromagnetic object is heated and reaches Tc the magnetization gradually drops as we get closer to Tc or it's a instant drop? Can I assume as I heat the object, the magnetization is weakening ...
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6answers
2k views

Why are Only Real Things Measurable?

Why can't we measure imaginary numbers? I mean, we can take the projection of a complex wave to be the "viewable" part, so why are imaginary numbers given this immeasurable descriptor? Namely with ...
4
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2answers
405 views

Adjoint of a Wave Function

Why is the adjoint of a function simply it's complex conjugate? Normally with a vector we consider the adjoint to be the transpose (And the conjugate? I don't know why), so does this concept carry ...
0
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1answer
481 views

Why is momentum (instead of something else) the canonical conjugate of position?

Why did nature decide to make conjugate of position to be momentum? Since energy and position do not commute, why not energy? What determines the pairing of time with energy and momentum with position?...
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0answers
60 views

Adiabatic evolution for initial Hamiltonian on Hadamard basis and problem Hamiltonian as diagonal

This is spawned from a comment at the answer to one of my previous questions. Someone suggested to me that claiming the following statement might be NP-hard. Could anyone please help me to figure out ...
5
votes
1answer
104 views

Classical EM neglects electron recoil?

Imagine two electrons $A$ and $B$ at rest. Electron $B$ is at a vertical distance $r$ above electron $A$. Let us assume that the electrons are constrained to move on horizontal rails. At time $t=0$ ...
1
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1answer
497 views

Normal Coordinates for Quantum Coupled Oscillators

Thanks if you take the time to read this. Here is the problem statement: The problem I'm getting is that I'm not getting the kinetic energy diagonal when I convert to the coordinates that ...
1
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1answer
128 views

Practical example of stabilizer codes

Given the Steane code $$ \left|0\right\rangle_L \equiv \frac{1}{\sqrt{8}}(\left|0000000\right\rangle + \left|1010101\right\rangle + \left|0110011\right\rangle + \left|1100110\right\rangle + \left|...
1
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1answer
75 views

Confusion about a lemma on the time constraint of an adiabatic evolution (arXiv:quant-ph/0604077)

I am going through the paper Quantum adiabatic evolutions that can't be used to design efficient algorithms by Zhaohui Wei and Mingsheng Ying. On the second page they prove a lemma. The statement goes ...
2
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1answer
256 views

Possible Outcomes from Measuring a Hydrogen Atom

A hydrogen atom is characterized by the wavefunction $$\mid \psi \rangle =\sqrt{\frac{2}{7}}\mid 4\,2\,1\rangle +\sqrt{\frac{1}{7}}\mid 2 \,1\,\bar{1}\rangle+\sqrt{\frac{4}{7}}\mid 3\,2\,0\rangle$$ ...
0
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0answers
81 views

Angular Momentum with Upper Index

I am asked to show $[L^2,L_i] = 0 $, but with the definition : $L^2 \equiv L_i L^i$ I tried this: $[L_i L^i,L_i] = L_i [L^i,L_i] + [L_i,L_i]L^i$ We know that : $[L_i,L_i]$ = 0 , so we have, $[L_i ...
0
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1answer
113 views

Momentum representation of a function with discontinuous derivative [closed]

Consider the following wave packet $$\psi = Ce^{2\pi i p_0x/h}e^{-|x|/(2\Delta x)}$$ where $h$ is the Planck's constant and $C$ is the normalization constant. The derivative of this function is ...
2
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2answers
458 views

Probability current vs. direction of wave function

I did an exercise for my Quantum-Mechanics Lecture: Let $\hbar$=2m=1. A particle in 1 dimension has $j(x)=2\ Im(\overline{\psi} (x) \ \psi'(x))$ and it's to show that there are superpositions $\psi (x)...
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2answers
61 views

Unitary Confirmation

I am asked to show that an new defined operator: $$U_{\beta} = \exp(\displaystyle\frac{i\beta L_z}{\hbar})$$ is unitary, where $$L_z = -i\hbar\,\,(x\displaystyle\frac{\partial}{\partial y} - y \frac{...
4
votes
3answers
1k views

Are matter waves transverse and can they be polarized?

Are matter waves transverse and can they be polarized? What I know:I'm aware of the de Broglie matter waves hypothesis and de Broglie wavelength relation(at a very basic level as part of high school ...
2
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0answers
131 views

Choice of X-ray scatterer in Compton effect

I am going to perform an experiment on Compton Scattering, and I am going to use an X-ray scatterer to scatter the incident X-rays. I have been instructed that Acrylic Glass slab are the best for this ...
5
votes
2answers
211 views

Do we need to expand the potential in a power series to show $[x, V(x)] = 0$?

Today in class (Intro to QM) we went over a couple of commutators. Among them was $[x, V]$, where $V=V(x)$ is a potential. What the teacher said to prove this is zero was: let's assume $V$ is analytic ...
2
votes
0answers
404 views

Theoretical or experimental violations of the 2nd Law of Thermodynamics? [closed]

Theoretical challenges to the 2nd Law? What are some the theoretical challenges to the 2nd Law? (cf. Čápek, Vladislav, and Daniel P. Sheehan. Challenges to the Second Law of Thermodynamics: Theory ...
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1answer
70 views

Simple photon recoil question

Imagine two charges A and B separated by some distance. Charge A emits a photon which is absorbed by charge B. Is the recoil momentum received by charge A always equal and opposite to the momentum ...
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1answer
157 views

Do electrons have structure in quantum gravity?

I have read Luboš Motl's blog and he says in QG electrons must be composite and have structure because QG says there can be no point like particles and everything above the Planck size can be resolved ...
2
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0answers
142 views

How to calculate angular momentum per second for a photon beam? [closed]

Given a photon polarization state $$|\phi\rangle = \frac{3}{5}|x\rangle + \frac{4i}{5}|y\rangle,$$ a beam of photons transmit $N$ photons per second in such a state. An L-polarized photon has an ...
4
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2answers
650 views

Understanding the Wave Function and Excited States

A wave function is an infinite dimensional vector space, how can it "live" in $\mathbb{R}^3$? Given the equation that is built like: $$\Psi (x,t) = \sum ^{\infty} _{n=1} c_n \psi _n (x) e^{-i E_n t / ...
5
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1answer
2k views

Orthonormality of Radial Wave Function

Is the radial component $R_{n\ell}$ of the hydrogen wavefunction orthonormal? Doing out one of the integrals, I find that $$\int_0^{\infty} R_{10}R_{21}~r^2dr ~\neq~0$$ However, the link below says ...
1
vote
1answer
86 views

Simple QM question about Sy matrix

Given a spin 1/2 particle in state $|\alpha\rangle=\begin{bmatrix}a \\b\end{bmatrix}$, what is the probability of it being measured in the $S_{y+}$ state. Is this equivalent to, if $S_y$ is measured ...
2
votes
1answer
924 views

Wave function collapse

When we try to measure the position of a system, wavefunction collapses to form a spike. After a while, the wavefunction spreads again, and you take another measurement, the results will be different" ...
13
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1answer
574 views

Operator that describes particle detector

In non-relativistic QM, the position of a particle is an observable. In QFT, fields are the observables. However, particles must have some sort of position, otherwise we wouldn't see pictures like the ...
2
votes
2answers
121 views

Quantum uncertainty in cell functions

In class today (philosophy of the mind) we discussed the ideas of Richard Lewontin. He stated that in determining the phenotype of a gene we must take into account the environment but also quantum ...
1
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1answer
117 views

Weird Behaviour of the act of measurement to a quantum system

I and my friend were disputing about some weird behaviour of the act of measuring some observables quantities e.g. Energy, position. But I still don't think what he said is strictly true. He said" ...
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1answer
2k views

Writing a Wavefunction as a Linear Combination of Eigenstates

We have the following wavefunction for the hydrogen atom: $$\psi(r,\theta,\phi)=\frac{1}{\sqrt{4\pi}}\frac{1}{(2a)^{3/2}}\frac{r}{a}e^{-r/2a}\sin(\theta)\sin(\phi)$$ where $a$ is the Bohr radius. <...
2
votes
2answers
196 views

Angular momentum matrices (Schiff section 27)

On page 203 3rd edition of Schiff we are given the angular momentum matrices ${J}$ for $j=1$. I am curious as to how these relate to orbital angular momentum for $j = 1$. If we take the corresponding ...
4
votes
2answers
698 views

How much time does it takes an electron to tunnel through a barrier?

I know that in quantum mechanics there is no "time operator", so such a question is ill-posed. Anyway if the tunneling is instantaneous, this would imply an information transmission faster than $c$. ...
2
votes
1answer
73 views

Purposes of QEC stabilizers

I am going through the idea of stabilizer formalism. Defined what is a Pauli group $P_n$ and its properties, we describe a stabilizer set $S$ as: $$S\subset P_n$$ The stabilizer set establishes ...
2
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0answers
97 views

Ehrenfest's theorem on Gaussians

Considering the free evolution of a Gaussian wave packet, is it possible to use Ehrenfest's theorem to determine the average value of momentum given that of position? And I imply the simplified ...
4
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1answer
216 views

Correct way to do a Thomas-Fermi approximation for cold gases

I have calculated the total Gross-Pitaevskii energy for a 2D Bose-Einstein condensate in an harmonical trap, using a variational gaussian wave function with a variational parameter b. Now I want to ...
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2answers
1k views

What is the significance of electron spin quantum number?

Somewhere I read that spin quantum number is a particularly interesting theory of quantum mechanics as what it really implies is that particles like electrons do not come back to the initial state of ...
3
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0answers
87 views

How many worlds does the world split into in the Many Worlds Interpretation? [duplicate]

I've been reading up on the Many Worlds Interpretation of Quantum Mechanics, and there is one thing (among many) that I really don't understand. How many worlds are 'created' by an 'observation' or '...
2
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1answer
1k views

Can two photons annihilate?

This is a question about definitions. When two photons interact to create an electron/positron pair, does this process 'count' as annihilation of the photons? I've struggled to find a good ...
3
votes
2answers
102 views

Identity in quantum operator tutorial

I'm reading this tutorial by Ben Simons entitled Operator methods in quantum mechanics in connection with his course in advanced QM, and I'm a bit puzzled by an identity in page 25, a bit above ...