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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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Have mass position schemes like the Arthurs and Kelly schemes been realised physically?

Basically, my question is whether there are methods to monitor mass position with an interaction Hamiltonian followed by a measurement of the detector wave function, as in an Arthurs and Kelly scheme. ...
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Dependence of wave function with time, especially probability density function. And Continuity equation

I was learning Basic Quantum mechanics. I cam across the fluid equation in QM, which suggests $\Psi^*\Psi$ is probability density function. Consider the two statements below Probability will change ...
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Does the background shift affects the renormalization group equations?

In Section 21 of "Quantum Field theory" by Mark Srednicki, it is shown that there are two equivalent ways to get the quantum action of the shifted field $\phi'= \phi-\tilde{\phi}$, where $\phi$ is the ...
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What happens to a wavefunction upon measurement when there's degeneracy?

I'll use a hydrogen atom as an example. A hydrogen atom has multiple energy eigenstates for all but one of its energy levels. Suppose I measure a hydrogen atom to have an energy $E_n$ where $n > 1$....
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What are ' non-trivial transitions' in this context?

So I am reading this paper where the goal is to prove that thermal states are the only completely passive states. To be more exact first the paper consider a single system with let's say a density ...
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Why must I solve the de Broglie relationship in a single dimension instead of all three?

Context: a particle of mass $m$ can move in 3D and is trapped inside of a sphere of radius $R$ and impenetrable walls (in a more mathematical sense, the potential energy is 0 inside of the sphere and $...
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Trying to first understand position and momentum bases in Quantum Mechanics

In my lectures, I am told: $$\langle x \mid \psi \rangle = \psi (x)$$ Which can only be valid if the overlap integral is: $$\langle x \mid \psi \rangle = \int_{-\infty}^{\infty} \delta (x-x') \ \...
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How does an outer product really work?

I'm taught that, for arbitrary wavefunctions $\psi, \phi$, that: $$\hat B = |\psi \rangle \langle \phi \mid$$ Which produces a new ket apparently when applied to a ket, as.. $$\hat B \ | \mu \...
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Schrödinger Equation for a freely falling body near the surface of Earth

Near Earth's surface the Schrödinger equation of a freely falling particle takes the form, $$ \frac {-\hbar^2}{2m} \frac {d^2 \psi (y)}{dy^2} + mgy\psi (y) = E \psi (y). $$ Putting $k=\frac {\sqrt {...
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‘Supersymmetrizing’ an arbitrary quantum-mechanical potential

To my understanding, it is not possible to $``\text{supersymmetrize}"$ an arbitrary quantum-mechanical system unless one knows how to represent the corresponding Hamiltonian in the form $$ H = A^\...
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Why is Quantized Inertia considered pseude-science? [closed]

In the criticism section of the Wikipedia article to Quantized Intertia it is stated that QI is considered pseudo-science. Unfortunately none of the linked but in none of the provided links can I find ...
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Does the dipole moment of an atom modelled as a two level system depends on its frequency?

Consider an atom modelled as a two level system : $$H=\frac{\hbar \omega}{2} \sigma_z $$ $|0\rangle$ and $|1\rangle$ are the ground and excited states that span the Hilbert space. In the Rabi ...
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Tensor product of kets and bras

My question is about tensor products. I have learned that the tensor product between two operators is, for example, $A{\otimes}B$. Suppose we have a system $A$ and a system B. Why we can write the bra ...
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How much information about a quantum operator is determined by its Poisson bracket Lie algebra?

Hamiltonian quantum mechanics is often built using many ideas from Hamiltonian classical mechanics like the Poisson bracket to determine the commutator between quantum operators, which is appropriate ...
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A confusion about why can't a statistical mixture be modelled as a superposition of pure states?

I have read Cohen's book, and various posts in this site; however, I'm still not convinced why we can't model a statistical mixture as a superpositions of pure states ? For example, consider the ...
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Definition of Entanglement

The definition of quantum entanglement, found on the internet and the literature is: On a bipartite system $\mathcal{H}_A \otimes \mathcal{H}_B$, let $\rho$ be a mixed state. It is said to be ...
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Why in the BCS ground state the probability amplitudes are taken real?

In some references (see for example Ballentine ch. 18.5) the ground state of the BCS theory is assumed to be \begin{equation} |BCS\rangle = \prod_{\bf k} (u_{\bf k}+v_{\bf k}\hat{c}^{\dagger}_{\bf k,\...
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Conceptual understanding of operators in QM

Do operators in QM represent in some fashion the action of the measurement apparatus on a state being measured? Usually operators in QM are introduced as abstract transformations whose eigenvectors/...
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What does the “lowest order adiabatic approximation” mean?

I came across a concept "lowest order adiabatic approximation" in solid-state physics. I searched and found dozens of "lowest order adiabatic approximation". I think the "lowest order adiabatic ...
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Exponential of the Pauli matrices [closed]

My job is to prove: $$\exp(i\theta \vec{v} \cdot \vec{ \sigma })=\cos(\theta)I+i\sin(\theta)\vec{v} \cdot \vec{ \sigma }$$ where $\theta \in \mathbb{R}$ and $\vec{v} \cdot \vec{ \sigma }=\Sigma^3_{i=...
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Probability of transition defined as the the ratio between reflected and incident fluxes

I'm reading a paper (Rapp, 1968) that treats quantum mechanically the problem of a particle $A$ "hitting" an harmonic oscillator made of two particles, $B$ and $C$: $\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,...
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Spin of 3 particles

I am trying to decompose the isospins of a three particle state using Clebsch-Gordan coefficients such as: $|1,1\rangle \otimes |1/2,-1/2\rangle \otimes |1,0\rangle$ Decomposing the first two ...
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How does time-translation symmetry morph into evolution in time?

I am reading Ballentine's textbook "Quantum Mechanics: A Modern Development". In it he transitions from discussing time-symmetry to discussing evolution (of the state) in time. I'm finding it ...
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Why is information indestructible in quantum mechanics? [duplicate]

Why is information indestructible in quantum mechanics?
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1answer
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Entanglement in quantum physics [duplicate]

Mathematically what is the difference between pure separable state and entangled state ? Can anyone explain with equations?
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30 views

Is a photon emitted from Earth less energetic that a photon emitted from the Moon

Two similar looking photons (locally) are emitted, one from Earth and the other from the Moon, and they are observed at some point X out in space. On Earth, time is slightly more dilated due to ...
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How to get the Number Phase Uncertainty relation from the Energy time relation?

I can arrive at $\Delta H\Delta t \geq \frac{\hbar}{2}$, but how do I get from there to $\Delta N\Delta \phi \geq 1$ for the number states of light? I know we write $H = \hbar \omega (N + \frac{1}{2})...
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Mathematical analysis of recombining streams in a Stern-Gerlach experiment

My quantum mechanics textbook skips some steps in its mathematical analysis of a Stern-Gerlach experiment, and I am having trouble filling in the blanks. The experiment sends a streams of electrons ...
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Momentum Change from Spontaneous Emission

In page 188 of Christopher Foot's book on atomic physics, he talks about the Doppler cooling limit, and how it comes from a random walk caused by spontaneous emission. He claims that each spontaneous ...
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Finding the uncertainty of coherent states using operators [duplicate]

The coherent state is defined such that $a|\alpha\rangle =\alpha|\alpha\rangle $. We can calculate the uncertainty using $$\sqrt{\langle x^2\rangle-\langle x\rangle ^2}\sqrt{\langle p^2\rangle-\...
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107 views

Does Bell's theorem imply nonlocality using a false assumption?

In https://arxiv.org/abs/1409.5158, the author concludes that Bell tests cannot refute local realism, because they employ a wrong analysis. He says: "The quantum joint prediction cannot be ...
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Commutator $\vec{L}$ with $\vec{X}\cdot\vec{P}$

Let $\vec{X}=(X_1,X_2,X_3)^T$ and $\vec{P}=(P_1,P_2,P_3)^T$. Define $\vec{L}=\vec{X}\times\vec{P}$. Then, I can calculate $\vec{L}=(X_2P_3-X_3P_2,\,X_3P_2-X_2P_3,\,X_1P_2-P_1X_2)^t$. For all ...
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Total charge in electron cloud in a hydrogen atom

Essentially, the problem I'm trying to solve is The potential at a distance $r$ from the nucleus that is generated by an electron in a hydrogen atom is given by: $$V(r) = \frac{q}{4\pi\...
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Can electrons “build up” energy?

We're studying the photoelectric effect. My book says that according to the classical wave model, there is no reason for electron emissions to occur almost right after turning on the light source. It ...
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How does a vanishing $[x, p]$ work with the group theoretical definition of $p \propto \frac{\partial}{\partial x}$?

Thought about this while I was looking at some stuff on quantum-classical correspondence and where precisely the difference between quantum and classical comes from. Usually it's said that the key/...
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QM probability to go from a point to another (Zee)

In Zee's QFT in a Nutshell book at p.10-11 it is piecewise said that : In quantum mechanics, for a Hamiltonian $\hat{H}=\hat{p}^2/2m$, the amplitude to propagate from a point $q_j$ to a point $q_{j+1}$...
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Problem on measurement of spin of an electron

I came across a problem which reads: "An electron is initially found to have z-component of spin=+h/4π. Then a measurement of component of its spin along x-direction is carried out but the result is ...
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Why does the triplet state $\dfrac{1}{\sqrt{2}}(\uparrow\downarrow+\downarrow\uparrow)$ have spin 1 and not 0?

Don't the spins in the state $\dfrac{1}{\sqrt{2}}(\uparrow\downarrow+\downarrow\uparrow)$ cancel each other so that the total spin is 0 just like for the singlet state $\dfrac{1}{\sqrt{2}}(\uparrow\...
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Particle Double slit

How was the Particle Double Slit Experiment really done? To be precise, what kind of double slits were used and how can such slits be small enough, especially for electrons and buckys balls?
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How come the xenon $6s$ state is metastable?

I'm having some trouble puzzling out some aspects of the electronic excitation spectrum of xenon, and I'd appreciate some help with it. This classic paper ─ the first one to pioneer the use of laser-...
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Delayed Choice Quantum Eraser - what is erased?

I fail to see why everybody thinks that something is erased in the famous Delayed Choice Quantum Eraser experiment (see https://arxiv.org/pdf/quant-ph/9903047.pdf and https://en.wikipedia.org/wiki/...
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Meaning of time derivative of an operator

Today when my professor was deriving this equation: $$\frac{\mathrm d\langle A\rangle}{\mathrm dt}=\frac{i}{\hbar}\langle\left[H,\,A\right]\rangle+\left\langle\frac{\partial A}{\partial t}\right\...
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Integration of Outer Product of a Basis in Quantum Mechanics

I'm using Griffiths' Introduction to Quantum Mechanics (3rd ed., 2018), and have come across what, on the face of it, seems a fairly straightforward identity, but which I cannot justify to myself. It ...
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Can we derive Schrödinger equation from classical wave equation?

In classical mechanics wave equation is $$y=A\sin(kx-\omega t)$$ $y$=instantaneous displacement, $A$=maximum displacement, $\omega$=angular velocity, $x$=position of particle, $k$=wave number Now in ...
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Bloch sphere representation of an eigenvector

I'm trying to work through a problem that wants me to determine the Bloch sphere representation of the eigenvectors of $\sigma_{z}$. I'm working in bra-ket notation so these would be $\ v_{+} = |0\...
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What is difference between fermions and spins?

A spin model i.e. $H_s = \sum_i^{L-1} S_i^x\cdot S_{i+1}^x$ can be written in matrix form as following $$H_s = \big(S_1^x \otimes S_2^x \otimes I_3^2 \otimes I_4^2\otimes \cdots\otimes I_{L-1}^2\big)...
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Hamiltonian of quantum harmonic oscillator with $\psi(x)=\delta(x)$: comparison to classical mechanics

I was just reading the question Why can't $\psi(x)=\delta(x)$ in the case of a harmonic oscillator? The accepted answer says that $\psi(x)=\delta(x)$ is a mathematically valid state, though it's not ...
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Calculating the $g^{(2)}$ correlation function

I want to make sure I am doing this right. I am also lost on how to continue at the end I have to calculate $$g^{(2)}(0) = \frac{G^{(2)}(0)}{|G^{(1)}(0)|^2}$$ for the following states (1) $$\...
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What does a “pure” state mean in QM? [duplicate]

Question: In Quantum Mechanics, people use the word "pure state" for some states; however, what do they mean exactly ? Thoughts: I mean, a state is a vector in our vector (Hilbert) space, so in that ...
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“Commuting observables share common eigenstates”

I am struggling to find a precise definition of this line from my quantum mechanics textbook: If $[A,B] = 0$, then the operators commute, and "commuting operators share common eigenstates". This ...