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Hubbard Model Hamiltonian in matrix form using basis

By definition, $H_1$ is the matrix in the basis $\beta$ of $H$ restricted to the eigenspace $N=1$ (this is well defined since $H$ commutes with $N$ therefore they admit a simultaneously diagonal ...
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What is really a negative energy particle (and why is it different from an anti-particle)?

Anti particles can be thought of as negative energy particles traveling backwards in time. In effect their reversed direction of time means they can really be understood as positive energy particles ...
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1 vote

How is light as electromagnetic waves produced?

Light is alway produced by excited electrons dropping back to a lower energy state and releasing photons. The excitation of the electrons can be caused by other electromagnetic waves or electric ...
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What exactly do we observe in particle accelerators that we have named "particles"?

Do we observe dots on a position detector like in the double slit experiment? Of course we do. They are classical, relativistic BB pellets, for all intents and purposes, as they don't interfere with ...
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Do the canonical commutation relations have any connection to geometry?

Roughly speaking, the gauge potential is identical to connection where the gauge potential is related to amplitude (field variable). The commutation relation in quantum theory can be written as the ...
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1 vote

Assuming the photon has a moving mass, why current quantum mechanics is unable to prove or deny it?

This question has been beat to death here, a quick search should give you more details. So, in a nutshell: Non-relativistic quantum mechanics doesn't describe massless particles well. In a ...
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7 votes

Contradiction in my understanding of wavefunction in finite potential well

Imagine a perfectly elastic ball dropped vertically onto a flat surface. The ball heads for the point of lowest potential, ie the ground, but because of conservation of energy it bounces back to its ...
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1 vote

Confused about the scattering Operator in LSZ reduction formula

In Greiner's Field quantization book, Chapter 9 on the LSZ reduction formalism, he states $$S_{fi}=\langle q_1,...,q_m;\text{out}| p_1,...,p_m;\text{in}\rangle\tag{9.10}$$ This might be written ...
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Can you see on Miller's planet?

Neglecting any brightness from the low energy part of the blue-shifted CMB (which might be significant, as ProfRob points out), a surface temperature of 890 degrees Celsius corresponds to a red-orange ...
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-1 votes

Can you see on Miller's planet?

You could see perfectly well. If you didn't get fried you could use a lantern to enlight the stuff around you. You could hide in a deep cave to take shelter from the burning CMBR (not sure if that has ...
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Can you see on Miller's planet?

I think the answer is yes, but the illumination levels would be similar to a heavily overcast day on Earth. The CMB is a blackbody continuous spectrum, so even though the peak is blueshifted into the ...
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10 votes

Contradiction in my understanding of wavefunction in finite potential well

Even classically, particles with a fixed total energy spend more time near the turning points since this is where the motion is the slowest. The probably of finding the particle in a small region ...
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2 votes

How does string theory relate superposition and general relativity?

Theories that allow for superpositions of space being curved in different ways have all of their inconsistencies show up at high energies. The new degrees of freedom which come to the rescue in string ...
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1 vote

How does string theory relate superposition and general relativity?

So I know that in general relativity, superposition cannot be true. General Relativity is a classical theory, and is not quantized yet, its quantization is a matter of current research, and string ...
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The reasoning of the definition of $S$-matrix

As lurscher already said correctly, the idea is to choose an initial and a final time that is (in time) away from the actual time-dependent interaction in order to make sure that the interaction does ...
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Question on computation with the bra-ket-notation

For an interval $\Delta$ we can define $P(\Delta)$ in the position representation (using the bra-ket notation) as follows: $$ \langle x|P(\Delta) := \mathbf 1_{\Delta}(x) \, \langle x| \quad, \tag{1}$$...
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Which book should I start to know good understanding of light-matter interaction?

Here is a free course about "atom-light interaction" online:https://ocw.mit.edu/courses/8-421-atomic-and-optical-physics-i-spring-2014/ The teacher is Prof. Ketterle who win the Noble price ...
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Is the identity $\rho=\sum_m M_m\rho M^t_m$ possible for measurement operators with $\sum_m M_m^t M_m=I$?

Assuming $M_m$ here are Kraus operators, the identity is true iff they represent the identity channel, which is true iff each $M_m$ is proportional to the identity. See this related post on qc.SE for ...
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2 votes

Do-It-Yourself physics experiment

The first cyclotron was a tabletop device. Then again, a cyclotron requires a source of ions, I don't know whether that is doable as a home project. Also, I don't know how high of a vacuum is required....
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Separability of an Hamiltonian with spin

Yes, $[a,L_j]=0,[a,L^2]=L_j[a,L_j]+[a,L_j]L_j=0$ if $a$ is the scalar under rotation.
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2 votes

Why possibility for X-ray to excite inner electrons higher than outer electrons?

I was wondering this myself! I found part of the answer and thought I'd share it here for others. Yes, what the original poster had in mind is correct. The inner core electrons have a much higher ...
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1 vote
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Criterion for stationary density matrix

Assume that the (complex) Hilbert space $H$ is finite-dimensional. For two hermitian operators $A$, $B$ on $H$ it holds that $$[A,B]^\dagger = - [A,B]$$ and thus $$\forall \psi \in H:\, \left(\psi,[A,...
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Criterion for stationary density matrix

Choose a basis in which $H$ is diagonal and assume for its eigenvalues $H_i\neq 0$. Then $C:= [\rho,H]$ has matrix elements $$ C_{ij} = (H_i - H_j)\rho_{ij},$$ and so \begin{align}\mathrm{tr}([\rho,H]^...
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What is the best criterion to discern between classical and quantum physics?

This is a complicated question in general. One possible answer is to say that a system will be classical if you can conceptually access all its physical properties without changing them by measuring ...
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What is the best criterion to discern between classical and quantum physics?

If you can observe (aka measure) an object without substantially disturbing it, it's a classical object. If the object is so small that any type of observation does substantially disturb the object, ...
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1 vote
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Deriving a recursion relation for Clebsch-Gordan coefficients

Where you went wrong is comparing apples and oranges. Your superb text is right. By the time it utilized (3.369), it "translated" the unconventional (3.370) into the standard convention, (3....
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-2 votes

Curvature of Hilbert space

Hilbert space is infinite dimensional space, by default it is continuous, has no curvature, and extends indefinitely in all directions. It also lacks any edges where the space ends, or wraps around on ...
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What is path ordered product? Please explain

This is used, e.g., in Keldysh approach to non-equilibrium systems (see, e.g., Rammer and Smith for review): the operators are ordered along a contour/path in a complex plane. This can be easily ...
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2 votes

Does passing an electric current along a strip of metal submerged in saltwater cause anything?

Yes. Pumping current into the hull of a ship in moored storage has been used for decades to prevent the hull from corroding in constant contact with sea water. To accomplish this, a very large carbon ...
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Intuitive understanding of the derivation of the Rayleigh-Jeans law

I guess I am struggling with the 2nd argument which was there is no limit on the number of modes of vibrations that can excited? Why would there be no limit? Can someone please explain? This follows ...
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In Quantum Mechanics is it possible to apply time evolution operator to wavefunction?

This is correct. You can easily derive it from bra-ket notation by projection unto a position state, let $$ |\psi(t_0) \rangle = c_1|\psi_1\rangle + c_2 |\psi_2\rangle $$ We have $\hat U(t)|\psi_n\...
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4 votes

In Quantum Mechanics is it possible to apply time evolution operator to wavefunction?

Using kets, because the evolution operator is linear, we can write : $$|\psi(t) \rangle = \frac{1}{\sqrt 2}\left(e^{-iHt/\hbar} |\psi_1\rangle + e^{-iHt/\hbar}|\psi_2\rangle\right) = \frac{1}{\sqrt{2}}...
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1 vote
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Can the expectation value behave like a function?

Consider a harmonic oscillator with mass $m$ and frequency $\omega$, and the state : $$|\psi\rangle = \frac 1{\sqrt{2}}\left(|0\rangle + i|1\rangle\right)$$ In this state we have : \begin{align} \...
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Propagation of a wavefunction on a Riemannian sigma model

The configuration space for this theory has even coordinates $x$ and odd coordinates $\psi$ (it is $\Pi TX$ as a supermanifold). The Hilbert space is (the $L^2$ completion of ) the space of function ...
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1 vote

Nonvanishing expectation value lesser Green's function

In condensed matter many-body physics vacuum usually means a state filled up to the Fermi energy (such as the ground state of conduction band in a metal), i.e., $$ a_\mathbf{p}|0\rangle = 0 \text{ if ...
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3 votes
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How to second-quantize an operator if the field operator is a spinor

Let $\mathfrak h $ be the $1$-particle Hilbert space. Let $\mathcal O$ be an operator on $\mathfrak h$. The second-quantized version of this operator acts on an $k$-particle state $S_\nu(\phi_1\otimes\...
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Schrodinger equation of linear combination of quantum states

Yes, indeed. In fact, for a two-by-two Hamiltonian one could evaluate the operator $e^{-iHt}$ exactly and compare the result with the solution of the SE for the same Hamiltonian (when solving it as a ...
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Why are hidden variables hidden?

A particle stream streaks a corner and the particles are deflected in such a way that a periodic intensity distribution can be observed on an observation screen. Any attempt to observe the particles ...
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Understanding the solution of the infinite spherical well

There is a footnote in that edition that explains this apparent 'typo', which actually is not. Griffiths claims that $N$ is related to $n$ and $\mathcal{l}$, but in a quite complicated way for the ...
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Spectral flow in IQHE

To answer 1, let us consider the Hamiltonian $H_\Phi$ on page 53, and ignore $V(r,\theta)$ for now. Then you can easily check that the wavefunction in the lowest Landau level takes the form $$ e^{im\...
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Physical meaning of Transpose of an Operator in Quantum Mechanics?

Suppose a linear map $A:V\to V$ is represented in a basis ${\bf e}_n$ by the matrix ${A^n}_m$. This means that $A$ maps ${\bf e}_n$ to $$ A[{\bf e}_n] = {\bf e}_n{A^m}_n. $$ The map $A$ ...
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2 votes
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Why are hidden variables hidden?

Answer to question in 1st paragraph is "no". The term "hidden" here is not being used to signify "impossible to detect"; it is being used to signify "a physical ...
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1 vote
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Pure state vs mixed state in this example

Yes your reasoning is entirely correct. You would now describe the state at hand with the density operator $$ \rho = |c_1|^2 |\psi_1 \rangle \langle \psi_1| + |c_2|^2 |\psi_2 \rangle \langle \psi_2|$$ ...
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1 vote
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Question regarding exercise of dynamics of spin-1/2 system

You need to do everything explicitly in the basis $\{|+\rangle,|-\rangle\}$. So you have $$|\psi(t)\rangle=\begin{pmatrix} \psi_+(t) \\ \psi_-(t) \end{pmatrix} \quad\text{and}\quad |\tilde{\psi}(t)\...
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2 votes
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Probabilities of eigenfunctions

15+5+3=23. Probability must add to one. Note added: The ratios are $$ 1+ \frac 13 + \frac 1 5 = \frac{15+ 5+ 3}{15}= \frac {23}{15} $$
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3 votes

Confusion about the Wigner-Eckart theorem

In principle there exist tensor operators of arbitrary angular momentum/spin - if you consider that the angular momentum generators $L_i$ themselves are a vector operator, then e.g. $L_i$ in a spin-3/...
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1 vote
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Delta function: Intuitive way for boundary conditions

Let's integrate the Schrödinger equation from $x=-\epsilon$ to $x=+\epsilon $. We get: $\\$ $-\dfrac{\hbar^2}{2\,m}\,(\psi'(\epsilon)-\psi'(-\epsilon))+ V_0\,\psi(0) = E\,(\psi(\epsilon)-\psi(-\...
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3 votes
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Gell-Mann Low formula vs time independent perturbation

The main reason modern textbooks introduce the Gell-Mann-Low formula is that it leads to a very simple proof of the Feynman rules. The argument roughly goes like this: first, we write $$ U(t_1,t_2)=\...
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1 vote
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Series expansion of unitary operators in terms of other operators

The general principle at work here is made precise by Stone's theorem on one-parameter unitary groups. In short, if $U(\epsilon)$ is a strongly continuous$^\dagger$ family of unitary operators ...
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-1 votes

If the Many world interpretation is correct, is a single observer "privileged" in the experienced world?

What you ask is an open question for the Everettian (many worlds) interpretation of quantum mechanics. In physics we really don't like to talk about something like "consciousness", and leave ...
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