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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12
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5answers
3k views

Classical limit of quantum mechanics

I have heard that one can recover classical mechanics from quantum mechanics in the limit the $\hbar$ goes to zero. How can this be done? (Ideally, I would love to see something like: as $\hbar$ ...
1
vote
1answer
31 views

How to understand Preskill's argument for degeneration of eigenstates?

In his notes on topological quantum computation on page 18, Preskill uses the "commutator" $T_2^{-1}T_1^{-1}T_2T_1 = e^{-2 i \vartheta}$ to show that the eigenstates of $T_1$ are degenerate. But I don'...
2
votes
1answer
140 views

Feynman's derivation of the Schrödinger equation

I'm reading the following article: Feynman's derivation of the Schrödinger equation In this article, the autor claims that Feynman derivation of the Schrödinger equation was a key aspect of the ...
0
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1answer
57 views

Why is decay of states in QM exponential? [on hold]

If we consider a two level system, generally we see exponential decay from the excited state to the ground state. Why is this? Some assumption about the noise must yield this result, but what ...
1
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1answer
42 views

Quantum state after change of magnetic field

I have the following conditions: $\lvert\psi(0)\rangle=\lvert+\rangle_x=\frac{1}{\sqrt{2}}\lvert+\rangle+\frac{1}{\sqrt{2}}\lvert-\rangle$. So the state at $t=T$ is $\lvert\psi(t)\rangle=\frac{1}{\...
0
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1answer
49 views

Intuition for defining basis for Hamiltonian in momentum representation

I am going through Quantum information approach to the Ising model: Entanglement in chains of qubits by Stelmachovic et al. In Section A.4, the authors determines the eigenvalues and eigenstates of ...
0
votes
1answer
118 views

Question regarding NaCl equilibrium separation

So I am tutoring someone later and one of the problems is from Eisberg/Resnick Ch 12. The potential energy $V$ of NaCl can be described emperically by $$V = \frac{-e^2}{4\pi\epsilon_0 R}+Ae^{-R/\...
0
votes
1answer
47 views

Block Diagonal Matrix Shankar Quantum Page 45

On page 45 of Shankar's intro to qm (you can find a pdf of it online if you want) he says that a specific operator has a block diagonal form because when it operates on some element of an eigenspace ...
0
votes
1answer
40 views

Total chemical potential of electrons in pn junction

I am reading this page about electron energy concept terminology. I am trying to apply that for the pn junction in equilibrium below. Could anyone help me to see if I get it correctly? C = ...
0
votes
0answers
44 views

Transparent finite wells

If the transparency coefficient for a scattering particle in a finite potential well is $T=1$ then the energies are that of a particle in a infinite potential well. Why is this? Is this a ...
2
votes
1answer
127 views

How force get transmitted when a magnet attracts iron?

According to particle physics , every fundamental force has its force carrier particle. Photon is a force carrier particle of electromagnetic force but how does force gets transmitted when a iron is ...
-4
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0answers
45 views

Single Slit Experiment in Quantum Mechanics

In wave optics, with a single slit, there would be an interference pattern, which can be explained from Huygens principle, but in quantum mechanics, When we shot a particle, say electron, one at a ...
2
votes
2answers
352 views

Commuting operators and Direct product spaces

Under what conditions is the common eigenspace of two commuting hermitian operators isomorphic to the direct product of their individual eigenspaces? When can an eigenket $|\lambda$1$\lambda$2$\rangle$...
3
votes
1answer
437 views

How does an operator transform under time reversal?

We know that a time-reversal operator $T$ can be represented as $$T=UK$$ where $U$ is some unitary operator and $K$ is the complex conjugation operator. Then under time-reversal operation, a quantum ...
22
votes
6answers
2k views

Is there something behind non-commuting observables?

Consider a quantum system described by the Hilbert space $\mathcal{H}$ and consider $A,B\in \mathcal{L}(\mathcal{H},\mathcal{H})$ to be observables. If those observables do not commute there's no ...
0
votes
0answers
27 views

How to do partial trace of three qubit? [on hold]

Good day, $\|A\rangle=\left(\dfrac{i_0}{j_1}\right)$, $\|B\rangle=\left(\dfrac{i_0}{j_1}\right)$, $\|C\rangle=\left(\frac{i_0}{j_1}\right)$, For 2-qubit systems, the $\|AB\rangle\langle AB|$, ...
2
votes
0answers
21 views

possibility of interference of electrons during its transition from higher to lower state

They say an electron possesses dual nature (what we call wave-particle duality in order to relate with our everyday world). If it is an electron (definite particle) it too shows wave-like phenomenon ...
2
votes
1answer
56 views

How to understand permutations of particles in Quantum Mechanics?

I'm studying identical particles in Quantum Mechanics and I'm having a hard time to understand the idea of permutations of particles from a mathematical standpoint. From one intuitive point of view ...
1
vote
1answer
106 views

Direct Sum representation of multiple particles in Quantum Mechanics

Suppose that I have three non-interacting spin-1/2 particles such that I can represent the combined system in a basis of \begin{align} D^{(1/2)}_1 \otimes D^{(1/2)}_2 \otimes D^{(1/2)}_3 & =(D^{(...
6
votes
1answer
223 views

Emergence of space from quantum mechanics

Once talking to a visiting professor at my institute, I heard about some simple model that captures the emergence of space coordinates as the eigenvalues of some infinite-dimensional quantum ...
6
votes
1answer
191 views

Uncertainty relation for non-simultaneous observation

Heisenberg's uncertainty relation in the Robertson-Schroedinger formulation is written as, $$\sigma_A^2 \sigma_B^2 \geq |\frac{1}{2} \langle\{\hat A, \hat B\}\rangle -\langle \hat A\rangle\langle \...
0
votes
1answer
37 views

If I prepare a state (density matrix) in Gibbs state, does it go to the lowest energy state at very low temperature?

Gibbs state is $$\rho_G=\frac{1}{Z} e^{-H/kT} = \sum_n \frac{1}{Z} e^{-E_n/kT}|E_n⟩⟨E_n| \, . $$ If $T$ goes to zero, does it mean the $\rho_G$ goes to the lowest energy state $|E_0⟩⟨E_0|$?
0
votes
1answer
25 views

How to prove that Gibbs state remains a gibbs state after evolution?

Gibbs state is $$ \rho_G=\frac 1Z e^{-H/kT}=\sum_n \frac 1Z e^{-E_n/kT}|E_n⟩⟨E_n|. $$. In wikipedia, it is said that a Gibbs state is an equilibrium probability distribution which remains invariant ...
0
votes
1answer
31 views

Normalisation of angular wave function: particle in a circular box

For a particle in a circular box (with radius $R$) with zero potential inside the circle and infinitely high potential outside of the circle, the Schrödinger equation in polar coordinates is: $$-\...
3
votes
2answers
59 views

Connection between harmonic potential and particle intepretation

I just finished a quantum mechanics course, but I still have some problems. In the simple harmonics potential well, energy between two adjacent states is always $\hbar\omega$. I read that this can be ...
0
votes
1answer
78 views

Why Doesn't Einstein Get More Credit for Being the Father of Quantum Theory? [on hold]

I'm not simply referring to the notion that Einstein treated the discrete emission and transference of energy (and matter) as "real" physical phenomena, but rather his major continuous role in the ...
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votes
0answers
30 views

Did de Broglie deduce E=pc? [on hold]

IIUC, $E=mc^2$ follows from $E=pc$ if $p=mc$ and $pc=hf$ is the same equation as the deBroglie wavelenth. Is it safe to say that $E=pc$ is an equation by de Broglie or is that equation some other ...
0
votes
0answers
22 views

Existence of eigenstates in type III staggered semiconductor heterojunctions?

Two semiconductors are aligned in the type III staggered fashion and sandwiched in between infinite potentials. Can there be an eigenstate across the valence band of one material and the conduction ...
0
votes
1answer
186 views

What is the form of the kinetic energy operator on a one-dimensional (real space) lattice? (In second quantization)

I'm trying to figure out how one would write down the Hamiltonian of a free fermion system (eventually in second quantization) on a one dimensional lattice and I'm having trouble both coming up with ...
0
votes
1answer
68 views

What is a Gibbs state?

What is a Gibbs state and what does it differ from a pure state? Say I have a two-level atom and it is described by a Gibbs state $\rho_G = \dfrac{e^{- \dfrac{H}{kT}}}{Z}$. I know $Z$ is a partition ...
0
votes
3answers
327 views

Einstein and vibrational energy of the atom and its way to QM

As suggested by one of the commentators on my last question, I am going through Bohr's Nobel prize lecture in order to understand how quantum mechanics was developed. The lecture describes Planck's ...
0
votes
0answers
17 views

Is the bandgap energy at the same level quantasized as other orbits of electrons?

Usually to excite an electron to an 'higher orbit' there has to be an exact quantity of energy of a photon. I don't know hów exact this quantity of photonenergy has to be, but is there a difference ...
0
votes
1answer
59 views

Density of states from $k$ to $E$

Speaking about Quantum mechanics, considering the "particle in a box" condition as an approximation of the electrons condition in a semiconductor, let the material be represented by a volume $V$ with ...
0
votes
1answer
170 views

Expectation value of total angular momentum $\langle J \rangle$

[I am working with Griffiths Introduction to Quantum Mechanics, 3rd Edition. My problem is general but if you want to look I am reading from ch 4.1 in which the weak-field Zeeman Effect is being ...
11
votes
1answer
196 views
+50

Significance of the exception to Gleason's Theorem when n = 2

Gleason's Theorem famously asserts that (appropriately defined) measures on the lattice of a complex Hilbert space can be implemented by density operators via the trace operation, except in the case ...
3
votes
1answer
116 views

A question on the Chern number and the winding number?

Let $\mid \psi(x,y) \rangle$ be a normalized wavefunction living in a $d$-dimensional Hilbert space and depend on two real parameters $(x,y)$ that belong to a closed surface (e.g., $S^2, T^2$, ...). ...
1
vote
0answers
25 views

Can an azimuthally symmetric perturbation lift the 2l+1 degeneracy of angular momentum eigenstates?

Assume the initial Hamiltonian of a spinless, non relativistic particle is $$H_0(r,\theta,\phi)=\frac{{\bf p}^2}{2m}+V_0(r)$$ Such that the eigenstates are angular momentum eigenstates $|n,l,m>$, ...
0
votes
0answers
31 views

What does phase shift represent in a Mach-Zehnder interferometer?

When describing the polarization of a photon, phase refers to the difference in the phases of the polarization components. E.g.: $|\psi\rangle$ = $|L\rangle$ + $i|R\rangle$ can be said to have a ...
0
votes
0answers
24 views

Mathematica code for self-consistent field loop in Hartree-Fock method

I want to write a Mathematica code for Hartree-Fock treatment of helium (or hydrogen molecule-> namely two electrons systems) , but I don't know how to write a loop for self consistent field procedure....
0
votes
1answer
42 views

Want to measure entanglement of the state [on hold]

Good day, I want to measure the state with concurrence and negativity. I do local unitary transformation with represented by $U\in SU(4)$ (Lie group). After the transformation (rotation of angle) ...
-1
votes
0answers
45 views

How to understand the momentum operator in a exponential function? [on hold]

What's the integral expression or matrix expression of the quantum parlance $$U\left( {x - \Delta {x_i}} \right) = \left\langle {x|\exp \left( { - i\Delta {x_i}P} \right)|U} \right\rangle ,$$ where $...
0
votes
0answers
20 views

Is this interpretation of the change in mass of a wave function correct?

I was interested in how a wave function of a single particle in no potential behaves if it was to lose mass. $$\psi = Ce^{ i \sqrt{2mE} x/\hbar}$$ So I took the derivative with respect to the mass, ...
1
vote
1answer
82 views

Probability in QM: derivation or interpretation? [duplicate]

It is known that coordinates $C_k\in\mathbb{C}$ of the QM-state vectors $|\psi\rangle$ has an interpretation as probability weights $p_k$ in the whole state through the formula like $|C_k|^2=p_k$. We ...
-3
votes
0answers
58 views

can mixed quantum states be thought as separable states

A quantum system contained 6 qubits is defined with Hamiltonian $H$, and has an energy spectrum of 36 energy levels. As statistical mechanics point of view, we can ...
-3
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0answers
25 views

3D wave equation of a 3D object [on hold]

For an example I have to derive wave equation of a sphere, x^2 +y^2 + z^2 = r^2 and its solution (wave function). Boundary condition can be anything. I like to know about eigenvalue and its use at ...
2
votes
0answers
32 views

Bosonization for unequal left/right Fermi velocities

The standard exposition of bosonization/Luttinger liquid theory in textbooks treats the case that left and right channels share the same absolute value of Fermi velocity. Is it possible to relax this ...
1
vote
2answers
88 views

How to form the matrix representation of $|O|^3$

I'm interested in getting the matrix representation of the absolute value of an operator. I know the matrix representation of the operator $O$. Now how do I take its absolute value?