Questions tagged [quantum-mechanics]

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-theory tag for the theory of many-body quantum-mechanical systems.

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Ladder operators for fermionic Fock space

To describe multiple fermionic particles, we introduce a Fock space $$\mathcal H_F=V_{\alpha=1}\otimes V_{\alpha=2}\otimes \ ...$$ such that each $V_\alpha$ is a two dimensional vector space labelled ...
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Position eigenstates in curved space

How does one define position eigenstates in curved space (say a manifold $\mathcal{M}$)? Let us say that it is defined as usual $$\hat{x}|x\rangle = x|x\rangle$$ Then how does one define the identity ...
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What is the role of the magnetic moment on wave particle duality

An electron, travelling at high speed relative to an observer, does not radiate unless it undergoes some form of acceleration. Yet we can observe wave like properties under certain measurement ...
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Can measurements be used for quantum communication?

My understanding of quantum entanglement is that when you measure the state of an entangled particle, its counterpart will measure a correlated state, i.e. we know for sure that if for example ...
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Infinite 3D sphere well with Dirac Delta potential function at the origin

A spinless particle of mass $m$ is constrained in a 3D region of zero potential within an impenetrable spherical shell of inner radius $r = a$, with a delta function potential at the origin given that ...
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How does the wave function relate to probability?

I'm trying to solve this problem which involves the probability of a particle being in a certain region. I know that $|\Psi|^2$ is the probability density, but how do I get this in the region?
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Truncation of BBGKY hierarchy

I have a (probably silly) question in expressing the first order equation of BBGKY hierarchy. I was able to obtain the following expression: \begin{gather} i\frac{\partial}{\partial t}\rho(1;1')=\bigg(...
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72 views

Is QFT “more” non-local than QM, at least mathematically?

Could physics still be local? Here's what I mean: The Schrodinger/Dirac equations allow for quantum entanglement, right? So in that sense they are non-local physically. But they are mathematically ...
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Can you put a direct sum in the exponent to represent the multiplicity of irreducible representations?

It is well known that one can decompose tensor product spaces into direct sums and that sometimes the components of the direct sums are not unique. Taking Wikipedia's example, the addition of three ...
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Commutator of an operator with its Hermitian adjoint in linear quantum systems

I have a commutator of a single-mode photon field operator $\alpha$ with its Hermitian adjoint $\alpha^{\dagger}$. [$\alpha$, $\alpha^{\dagger}$] When this commutator gives a negative value, $\alpha$ ...
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Thermal average of this operator

Anyone know how I might work out what something like this $$\frac{1}{Z}Tr(e^{-\beta H_{B}} (\hat{a} + \hat{a}^{\dagger}) e^{V_{n}(\hat{a}^{\dagger} - \hat{a}) - V_{n'}(\hat{a}^{\dagger} - \hat{a})})$$ ...
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Why does symmetry transform of an operator form like this?

\begin{equation} \hat{P}_R\hat{L}_{\rho}^{k}\hat{P}_R^{\dagger}=\sum_{\lambda}\hat{L}_{\lambda}^{k}D_{\lambda\rho}^{k}(R) \end{equation} where $\hat{L}_{\rho}^{k}$ is an operator, $\Gamma_k$ is an ...
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Schrodinger equation with parameters

I need to know the ground state energy $E_0$ defined by the following stationary Schrodinger equation: $$ -\frac{a}{2}\phi''(\xi) + \left(\frac1{2a}\sinh^2(2\xi) + (2b-1)\cosh(2\xi)\right)\phi(\xi) = ...
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What is the amplitude factor means?

There is a localized particle in a volume $V_0$, and we prepare a wave packet. It is written with eigenstates $ \chi_{a,q} $ as below. $$ X_{a,p}=\int a({\bf q}-{\bf p})\frac{e^{i{\bf q}\cdot r}}{(2\...
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Why can this integral be described as $\Gamma_{i^{\prime}} \otimes \Gamma_{j} \otimes \Gamma_{i}$?

Assume that the perturbation $\mathcal{H}^{\prime}$ is given as \begin{equation} \mathcal{H}^{\prime}=\sum_{j,\beta}f_{\beta}^{(j)}{\mathcal{H}^{\prime}_{\beta}}^{(j)} \end{equation} where $\beta$ ...
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Are there any explanations for Inertia at atomic level?

At macroscopic level we can observe inertia. But what explanations are there for Inertia at molecular/atomic/quantum level?
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Effects of a Lorentz boost on the normalization of a probability density (Dirac equation)

My question regards the probability densities of the Dirac equation. As is well known, the Dirac equation implies a continuity equation $$ \partial_\mu j^\mu = 0 $$ for $j^\mu = c\overline\psi \gamma^\...
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Possible energy states of atomic carbon

In Thornton and Rex's modern physics, chapter 8 (atomic physics), there is an example about the possible energy states of atomic carbon. Since the last two electrons are in the unfilled 2$p$ subshell, ...
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Explanation of quantum entanglement incomptabile with our intuitive human perception

I can't explain why the quantum entanglement can be demonstrated rationaly whereas our intuitive perception makes feel us this phenomena impossible to occur. Einstein has worked on it and tried to ...
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General wavefunction for a system of two coupled, quantum oscillators

Suppose we have two quantum harmonic oscillators, with different masses $m_{1},m_{2}$ and frequencies $\omega_{1,2}$. Then we can say particles are \emph{distinguishable}, in the sense that particle $...
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Shor's algorithm entanglement verification

I would like to ask whether the entanglement verification is necessary in Shor's algorithm In the paper, Nature Photon 6, 773–776 (2012), they mentioned that they tried to factorize 21 to avoid the ...
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Are there any 3 or more hermitian solutions to the problem: $\alpha_i^2=1$, $\{\alpha_i, \alpha_j \}=2$

I’m trying to generate some matrices which are similar to Pauli’s but with the following anti-commutation relation $$\{\alpha_i, \alpha_j\}=\alpha_i \alpha_j + \alpha_j \alpha_i = 2 \tag{1}$$ And $$\...
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Probability of finding particle in ground state [closed]

A particle is in a one dimensional infinitely deep square well, from 0 $\leq x \leq L$. Find the probability of finding the particle in the ground state of the square well if the wave function for a ...
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1answer
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Superposition of states of different fermion number

Physically, can quantum-states which are a superposition of states of different numbers of fermions exist? i.e. states of the form $\vert \psi \rangle = a \vert N\rangle + b \vert N' \rangle$ where $N ...
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Submerged shells, why are the valence electrons not in the outermost shell?

The valence shell of an atom, is the set of orbitals which are energetically accessible for accepting electrons to form chemical bonds. For various atoms, the valance electrons are submerged beneath ...
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How to map the eigenvectors of a superoperator into the corresponding operators?

The set of linear operators acting on a $d$ dimensional Hilbert space, $H$ form a vector space, called operator space $\mathcal{L}(H)$. Elements of operator space are $d \times d$ matrices. Now the ...
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Is the Larmor precession of $s$-orbital electrons constant, heedless of additional electronic orbitals as they get added from one element to another?

Understanding how the frequency required in NMR changes from element to element, even if all of them depend on the Larmor precession of the proton, I was wondering if they same applied to electron ...
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Proof of normalization constant of wave function to be independent of time

I am trying to prove that the normalization constant is independent of time. If we have fixed it for a particular time then it will remain constant for all time. Suppose $\psi(x,t)$ is a wavefunction. ...
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Reducing the number of parameters of a quantum state from 4 to 3

We have a quantum state $$ |\psi\rangle = \alpha|0\rangle + \beta|1\rangle, $$ where $\alpha$ and $\beta$ are complex numbers, i.e. $\alpha = a + bi$ and $\beta = c + di$. Therefore, our current ...
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Transmission Coefficient of Finite Square Well

Now I was calculating the Transmission Coefficients of finite square well potential and found something weird The transmission coefficient is given by $$T=\left[1+\frac{V_o^2\sin^2(2ka)}{4E(E+V_o)}\...
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How do I use the formula $|\langle \psi | \psi \rangle|^2$? [closed]

Suppose I have a state: $$ | \psi \rangle = \pmatrix{ a_1+ib_1\\a_2+ib_2} $$ I wish to calculate the probability of observing the state $\psi_1$. $$ \begin{align} \langle \psi_1 | \psi \rangle &= ...
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Quantum probabilities of a projection - does it require two different definitions…?

Say I have a wave-function $$ |\psi \rangle=\pmatrix{a_1+ib_1\\a_2+ib_2} $$ where of course $\langle \psi |\psi \rangle=1$. I can get the probability of a given state as follows: $$ \begin{align} \...
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Expansion with respect to a uncountable set of eigenvectors

Let $\mathscr{H}$ be a Hilbert space. If $\{e_{\alpha}\}_{\alpha \in I}$ is a Hilbert (orthonormal) basis, one can write every element $\psi \in \mathscr{H}$ as: \begin{eqnarray} \psi = \sum_{\alpha \...
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Inner product of $\langle \phi | \psi \rangle$ gives a complex value - why/meaning?

Say I have the following two wave-functions: $$ |\psi\rangle= \pmatrix{a_1+i b_1\\ a_2+ib_2 } $$ $$ |\phi\rangle= \pmatrix{c_1+i d_1\\ c_2+id_2 } $$ Since these are unit vectors of the Hilbert space, ...
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1answer
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Symmetry in solid state

I learned that a physical system with higher symmetry lowers the overall energy. If it is true, why we don't have many/all elements crytalized in simple cubic structure?
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Was the double slit experiment tried with slow light?

On wikipedia it is stated that light has been slowed to 17 m/s: In 1998, Danish physicist Lene Vestergaard Hau led a combined team from Harvard University and the Rowland Institute for Science which ...
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What is the eigenvalue of coulomb potential $V(\vec{r}_1-\vec{r}_2)=\dfrac{e^2}{\left|\vec{r}_1-\vec{r}_2\right|}$?

Assume that there is a comlomb potential between 2 electrons $V(\vec{r}_1-\vec{r}_2)=\dfrac{e^2}{\left|\vec{r}_1-\vec{r}_2\right|}$. In quantum mechanics, $V\to\hat{V}$, so $V$ will be $ V( \hat{\vec{...
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Quantum harmonic oscillators with momentum-position coupling

I have two coupled quantum harmonic oscillators given by the following Hamiltonian: $$H=\frac{p_{x}^{2}}{2}+\frac{\omega^{2} x^{2}}{2}+\frac{p_{y}^{2}}{2}+\frac{\Omega^{2} y^{2}}{2}+\frac{C p_{x} y}{2}...
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Color of pinholes of two different sized blackbody enclosures

Like this video shows, blackbody enclosures held at the same temperature and having the same dimensions, albeit made of different materials, show, as expected, the same color of the pinhole, despite ...
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Indistinguishability of Quantum States and its Consequences

In the book Quantum Computation and Quantum Information, there is a discussion about how if states are not orthonormal then there is no quantum measurement capable of distinguishing the states. I am ...
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How do I prove the identity for ${\rm tr}_p [e^{-iS\Delta t}(\rho\otimes\sigma)e^{iS\Delta t}]$ in Seth Lloyd's 2014 Quantum PCA Paper?

Equation (1) in Seth Lloyd's paper on Quantum PCA says: $\text{tr}_{p}\text{e}^{-iS\Delta t} \rho \otimes \sigma \text{e}^{iS\Delta t} = \cos^2(\Delta t)\sigma + \sin^2(\Delta t) \rho - i \sin(\Delta ...
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Showing that the Dirac adjoint satisfies the Dirac equation

If $\Psi$ is a solution of the Dirac equation then $\bar\Psi=\Psi^\dagger \gamma^0$ satisfies the equation i$\partial_{\mu}\bar\Psi\gamma^\mu$+$\bar\Psi\frac{m_{0}c}{h}=0$. How would I go about ...
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Can phonons induce non-$s$-wave superconductivity?

In BCS theory the interaction is considered isotropic and the result is an $s$-wave superconductor. Is it possible anyway for unconventional superconductors to have SC driven by phonons (I intend this ...
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My question is about the fractionalized electric charge in graphene, how to compute it i really got no idea? [closed]

my question is about the fractionalized electric charge in graphene, how to compute it i really got no idea?
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1answer
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Photon Catalysis

I am doing my research in the field of Quantum Optics. My topic of interest is Photon Catalysis. In this process, a coherent state and a fock state |1> are incident on the "a" and "b&...
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Creation and annihilation operators in coordinate space

I am trying to express the creation and annihilation operators of a single quantum harmonic oscillator in coordinate space. The problem is that, when I use $P \to -i\hbar d/dx$, I get $a=a^\dagger$: $$...
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Is energy really quantised? [duplicate]

I'm currently doing an introductory course to quantum mechanics, and came across an assumption that Planck used in solving the UV catastrophe. From what I understand, he essentially stated that that ...
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45 views

What are the rules for the principal quantum numbers of spin=1 particles? [closed]

I'm solving some practice questions and I found this one: Consider three particles with spins $S_1 = S_2 = S_3 = 1.$ (a) Find the dimension of the Hilbert space of this system. 27, since $(2S_1+1)(...
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CHSH Bell measurement, why is there difference between 45 and 135 degree?

Let assume BBO type-2 that simultaneously generates entangled photon pair in forms of horizontal polarization (H) and vertical polarization (V) of photon. One photon is going to Alice and the other is ...
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Aharanov-Bohm effect on a scattering system

I have a scattering system with a localized potential and a hard wall. Hardwall is at the x<=0. On the x-plane at an arbitrary point (x>0), there is a localized potential that scatters the ...

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