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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Converting between (abstract) linear operators and their position representations

Just as we have an abstract state vector $|\psi\rangle$ and its position representation $\psi(\vec{x}) = \langle \vec{x} | \psi \rangle$, how do we transform between a linear operator, say $H$, that ...
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14 views

Off-diagonal terms of the Husimi $Q$ function?

The Husimi $Q$ function of a quantum state $\rho $ is defined as $ Q (\alpha)=\langle \alpha \vert \rho \vert \alpha \rangle $, where $\alpha = (x, p) $ is a phase space coordinate and $\vert \alpha ...
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42 views

Can “state” be considered a 5th dimension?

I searched for an answer to this question on Google but just found articles that mention either string theory or a 5th dimension in passing (such as Maxwell equations as they relate to Riemann ...
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43 views

Quantum vector space - A complex vector space [duplicate]

Why is vector space of states, a complex vector space? and not a real vector space or perhaps a space based on a new field altogether, which we would have to create specifically for quantum mechanics? ...
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34 views

A meaningful distinction between determinism and causality

Causality is generally accepted to be a fundamental physical principle. But quantum mechanics is acausal (e.g. there is no 'why' as to the result of a measurement of the position of a particle in an ...
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1answer
31 views

Quantum entanglement on cosmological scales

This may be a foolish question given my limited understanding of QM but here it is. As I understand quantum entanglement basically means that two particles evolve as a single "unit", i.e., are ...
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0answers
17 views

How can I simulate a model electronic hole?

Suppose I can solve time-dependent Schrödinger equation for several 1D particles (currently 3). I'd like to see, what an electronic hole is and how it behaves — in a series of numerical experiments. ...
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26 views

Ground state of spin in magnetic field

I am trying to solve a time dependent perturbation theory problem, and it involves the Hamiltonian $$H=-\mu B\sigma_z$$ And a perturbation $$V=-\mu B_1\sigma\cdot(\cos(\omega t)\hat x-\sin(\omega ...
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23 views

Matrix forms to understand a state transition operator

Given an equilibrium of two states.$\left| 00 \right> \rightleftharpoons \left| 11 \right>$. And introduce a map $\mathcal{B}(\mathbb{C}^2\otimes \mathbb{C}^2) \rightarrow ...
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2answers
376 views

Why uncertainty principle is not like this?

In Griffiths' QM, he uses two inequalities (here numbered as $(1)$ and $(2)$) to prove the following general uncertainty principle: $$\sigma_A^2 \sigma_B^2\geq\left(\frac{1}{2i}\langle [\hat A ,\hat ...
4
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0answers
76 views

Degenerate perturbation theory applied to topological degeneracy?

Consider a quantum system described by a gapped Hamiltonian $H_0$ with degenerate ground states (GS), adding a perturbation term $V$ to $H_0$, then the low-energy physics can be described by an ...
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41 views

When can I use semiclassical approximation?

I know that I can use semiclassical approximation for path integral approach (in quantum mechanics) $\int d[q]e^{iA}$ when action $A >>1 $. But how shall I use such condition? For example, ...
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46 views

Heisenberg Uncertainity Principle

If any senior member of the group has access to the book, The Physical Principles of Quantum Theory by W. Heisenberg, then please help me in understanding the first section of chapter 2 where he gives ...
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44 views

Wave Function Integral I need help conceptually and Mathematically

$$\int_{-\infty}^{\infty}\frac{\partial^2\bar{\psi}}{\partial{x^2}}\frac{\partial\psi}{\partial{x}}~dx.$$ I have read that this is equal to Zero. Only problem is that what I am reading about doesn't ...
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1answer
33 views

Normalizing the sum of wavefunctions and calculating probabilty - understanding concepts

A state of a particle bounded by infinite potential walls at x=0 and x=L is described by a wave function $\psi = a\phi_1 + b\phi_2 $ where $\phi_i$ are the stationary states. So let's say we want to ...
3
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0answers
99 views

“Derivation” of the Heisenberg Uncertainty Principle

Ok, so I posted this in the mathematics StackExchange, but got no response. The question I outline below is my textbook's "derivation" of the Heisenberg Uncertainty Principle. The "derivation" my ...
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1answer
37 views

Does the energy-time uncertainty principle require energy levels to have finite width?

The uncertainty principle also has the form: $\Delta$$E$$\Delta$$t>h/2\pi$ Now this should mean that the thickness of the lines we draw in the energy level diagrams to show energy change undergone ...
3
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1answer
71 views

The momentum of a hole

I'm currently working through "A Guide to Feynman Diagrams in the Many-Body Problem" by R.D. Mattuck (self study, not a homework problem) and am stumped by the following problem: "In a system of free ...
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2answers
111 views

Can we describe mathematics using filters and matrices? [on hold]

Can quantum mechanics be partly explained in terms of mathematical filters? Is there a way to explain some of it with matrices on an amateur level?
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29 views

Energy in an electromagnetic wave

A radio antenna creates EM waves through switching the polarization in the antenna at a certain frequency. I assume the the energy of the photons produced in this process amount to E=hf for each ...
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0answers
15 views

Entropy of Reeh-Schlieder correlations

Any state analytic in energy (which includes most physical states since they have bounded energy) contains non-local correlations described by the Reeh-Schlieder theorem in AQFT. It is further shown ...
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54 views

Rewriting $\langle {\bf k} \vert E,l,m \rangle$ as $\langle {\bf k} \vert ~k,l,m \rangle$ Spherical Harmonics

From Sakurai eq. 6.4.21a we have that $$\langle {\bf k} \vert E,l,m \rangle=\frac{\hbar}{\sqrt{M k}}\delta\left(E-\frac{\hbar^2 k^2 }{2M}\right) Y_l^m({\bf\hat k}),$$ where $M$ is the mass of the ...
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1answer
65 views

Quantum Mechanics - Observable

If $O$ represents an operator corresponding to an observable why does the following equality hold? $$\langle f(x)\, |\, O g(x)\rangle = \langle g(x) \,|\, O f(x) \rangle$$ It is used on the last ...
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34 views

group and phase velocity of free particle [duplicate]

If Schrödinger wave equation is for matter waves then for a free particle Group velocity $V_g =2$ Phase velocity $V_p$ But matter waves satisfy the relation $V_g V_p = C^2$ where $V_p>C$ Does this ...
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1answer
25 views

Band structure and band index

Please let me know If my understanding is right. For a given $\vec{k}$, $H$ is a function of $\vec{k}$ the energies vary discretely for $n$ ie.,the band index. For a given $n$, we choose all the ...
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0answers
27 views

Density of States of Free Particle in One Dimensions

I am using Quantum Mechanics by David H. McIntyre, chapter. This is problem 15.7: Find the density of states $g(E)$ for the case of a free particle in one dimension; further, show that the density of ...
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0answers
15 views

A problem concerning the calculation of parameters in a periodic system [on hold]

I am using Quantum Mechanics by David H. McIntyre, chapter 15. The question is 15.8: Find the single bound state energy of an electron in an isolated well of depth $V_{0} = 1$ eV and with width $b = ...
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0answers
17 views

What are the New Researches in the field of laser physics? [on hold]

I want to know the newest updates of science in the laser physics researches specially the theoretical part of it.
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1answer
158 views

Can nowadays spin be described using path integrals?

In Feynmans book, "Quantum mechanics and Path Integrals" he writes in the conclusions (chapter 12-10) With regards to quantum mechanics, path integrals suffer most grievously from a serious ...
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76 views

$\psi^*$ if you have sine or cosine function [on hold]

If my $\psi$ function is $\sin(\pi x/L)$. What is $\psi^*$ going to be?
2
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0answers
62 views

Different hamiltonians for quantum harmonic oscillator?

The Hamiltonian for a classical simple harmonic oscillator is $$ H = \frac{p^2}{2m} + \frac{1}{2}m\omega^2x^2$$ With the usual choice of the ladder operators $$a = ...
2
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1answer
85 views

What kind of problem in quantum mechanics can have an algebraic method of solution?

For example, a harmonic oscillator can have an algebraic solution, and hydrogen potential can also have an algebraic solution. Here the algebraic method of solution means that we can use the similar ...
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1answer
49 views

First Order Correction to wave function in ground state

I am looking at a spin 1/2 particle in a magnetic field. This has Hamiltonian $$H=-\mu s\cdot B_0$$ For simplicity, assume $B_0=B_0\hat z$ so $H=-\mu B_0$. I then apply a perturbative magnetic field ...
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1answer
41 views

Perturbation of coupled spin

I am given a system with Hamiltonian (all 1/2 spins) $$H_0=\alpha(S_1\cdot S_2)$$ I broke it down and found that there were four eigenstates: $|1,[0,\pm1]\rangle$ and $|0,0\rangle$. Each has an ...
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44 views

Non-commuting vector multiplication in GR

If the tensors of GR were composed of non-commuting vector multiplications, this would at least in spirit bring GR closer to QM. Has this approach been attempted?
3
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1answer
51 views

Is obtaining the coordinate representation of momentum operator from commutator more fundamental than generator of translation

Related post: What is the most general expression for the coordinate representation of momentum operator? There are two methods of obtaining the coordinate representation of momentum in quantum ...
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26 views

Proving two forms of atom-field interaction perturbation Hamiltonian are equivalent

In the presence of an electromagnetic field in the dipole-approximation (${\boldsymbol A} = {\boldsymbol A}(0,t)$) we have the two forms $$H_{{\boldsymbol d}\cdot {\boldsymbol E}} = - q {\boldsymbol ...
4
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1answer
99 views

Time-Energy Uncertainty Principle and Operators

In most of examples, I notice that uncertainty principle for time & energy is given between mass & lifetime. The UP for time and energy is $$ \Delta t\,\Delta E\geq\frac h{4π} $$ where $$Δt ...
2
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2answers
165 views

From Quantum Mechanics to Classical Mechanics [duplicate]

Is it possible, and has it been attempted, to use quantum mechanics to deduce Newtonian, macroscopic level mechanics laws as was the case of statistical mechanics deriving thermodynamic relations?
2
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1answer
27 views

All angle dependence in $\mathrm{d}LIPS_2$?

Recall that $\mathrm{d}LIPS_2$ (one particle decaying into two particles of the same mass) is given by $$\mathrm{d}LIPS_2 = \frac{\vert{\bf k_1'}\vert}{16\pi^2\sqrt{s}}\mathrm{d}\Omega_{cm}.$$ In a ...
3
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1answer
96 views

Mandelstam variables 1 positive 2 negative

The three Mandelstam-variables are defined as: $$s=(p_A+p_B)^2=(p_C+p_D)^2,$$$$t=(p_A-p_C)^2=(p_B-p_D)^2$$$$u=(p_A-p_D)^2=(p_B-p_C)^2.$$ Where A and B are the incoming particles and C and D are the ...
3
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1answer
59 views

De Broglie Wavelengths

I have a working knowledge of wave-particle duality, I think. I know the de Broglie wavelength is a sort of probability of finding a particle in a specific position, and is calculated by ...
1
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1answer
72 views

Normalized Projection Operator

What is meant by normalized projection operator? What is its physical meaning in quantum mechanics? I am pretty confused regarding the physical interpretation of both projection operator and ...
3
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1answer
87 views

Naive questions on the ground states of Kitaev model

I got some naive questions on the ground states of honeycomb Kitaev model (with open boundary conditions): (1) Consider a simple case that $J_x=J_y=0$, then the model reduces to $$H=J_z\sum_{z\text{ ...
0
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1answer
27 views

Expectation value of Hamiltonian on number state [closed]

Hamiltonian is defined by $H_I = \hbar \omega (\hat{a}^+ \hat{a} + 1/2)$ What is the expectation value of the energy on the number state $$\vert \psi \rangle = \frac{1}{\sqrt{2}} ( \vert 1 \rangle ...
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1answer
41 views

Interpretation of Heisenberg's uncertainty principle

Heisenberg's uncertainty principle is one of the most fundamental principles on which quantum mechanics is based on. But it is also one of the most confusing laws we encounter. My doubt is whether the ...
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2answers
52 views

Measurement of Mass and Momentum of a particle simultaneously

In quantum mechanics can the mass and the linear momentum of a particle be measured precisely or do they commute ?
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1answer
64 views

How to derive the commutation relationship between $\hat{L}^2$ and $\hat{\textbf{p}}$ [closed]

How to prove that $$[\hat{L}^2,\hat{\textbf{p}}] = i\hbar(\hat{\textbf{p}}\times\hat{\textbf{L}} - \hat{\textbf{L}} \times \hat{\textbf{p}})$$ I tried to expand $\hat{L}^2$: ...
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1answer
31 views

Electronic configuration for singlet and triplet states

Is there a difference in the electronic configuration for singlet and triplet states? For example, He atom has 1s2 configuration in its ground state (singlet state) But what about when the He atom is ...
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0answers
26 views

Quantum Cryptography Open Problems [closed]

I am a 3rd year Physics undergraduate interested in Quantum Cryptography. This summer , I want to work on a open problem in Quantum Cryptography. I have credited courses such as Quantum Computation ...