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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Observables - what are they?
I often read in books that an observable is represented by an Hermitean operator. But it is deceiving as operator isn't the observable.
As far as I've read the observable is denoted like $\langle \...
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Proving that the electronic Schrödinger equation has no closed analytic solutions for >1 electron
It is stated in many books that analytic closed solutions to the time-independent electronic Schrödinger equation,
$$\hat{H}\Psi = E\Psi, $$
exist for the one-electron problem (e.g. hydrogen atom, ...
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Why are $SU(2)$-generators interpreted as *spatial* components of spin?
The generators of the unitary representation of $SU(2)$ on the internal spin Hilbert space of (say) a spin-$1/2$-particle are typically said to represent components of spin along various spatial axes. ...
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Randomness, Chaos, Quantum mechanical probability functions
Can someone explain these 3 concepts into a unified framework.
Randomness : Randomness as seen in a coin toss, where the system follows known and deterministic (at the length and scale and precision ...
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Web references for Nelson's "Quantum Fluctuations"?
Edward Nelson's book "Quantum Fluctuations" (Princeton UP, 1985) gives an alternative way to introduce trajectories, quite different to the trajectories of de Broglie-Bohm type approaches. I've read ...
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Is it really right to say that we never measure anything exactly in QM?
In reference to this elaborate answer by @DanielSank, I would like to pose the following question(s) in order to verify my understanding of the subject matter--in particular, that of the nature of ...
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Can I apply the Born rule to a Dirac spinor?
How does a Dirac spinor such as:
$$
\psi = \pmatrix{a_0+ib_0\\a_1+ib_1\\a_2+ib_2\\a_3+ib_3}
$$
Connect to a probability?
Can one apply the Born rule of this object?
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What are the similarities/differences between the behaviors of Quantum particles and bouncing droplets? [duplicate]
Bouncing droplets on a fluid surface show many weird behaviors of the quantum world. Look at this for example:
https://arxiv.org/abs/1307.6920
They can show tunneling, double-slit interference ...
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How long does an electron stay on a given orbital?
Was wondering what the average time is for an electron on any given orbital, or how often they change energy levels.
Thanks in advance.
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Tensor Product vs. Direct Product for three spin-1/2 particles
Let us consider three spin-1/2 particles and only focusing on their intrinsic spin $S$. The Hilbert space has then to be $\mathcal H = ℂ^2 ⊗ ℂ^2 ⊗ ℂ^2$. The spin can be described by $V ∈ \text{SU(2)}$ ...
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Distinguishability in Quantum Ensembles
Inspired by this question: Are these two quantum systems distinguishable? and discussion therein.
Given an ensemble of states, the randomness of a measurement outcome can be due to classical reasons (...
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What is the description of measurement in the Heisenberg picture?
In all the books I've read this picture is presented only briefly, by essentially saying that in the HP the whole time dependence is assigned to the operators (representing observables), whereas the ...
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Ehrenfest Theorem and boundary Conditions
In what cases does Ehrenfests Theorem hold?
If I look at the wavefunction of electrons in a squared box of length $L$ (with periodic boundary-conditions, $\Psi(0) = \Psi(L)$), then the solution to ...
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For a constant magnetic field, is there a gauge with both canonical momenta conserved?
To describe a constant magnetic field $\mathbf B=(0,0,B)$ (ignoring the motion along the $z$ dimension) within hamiltonian (or quantum) mechanics, one needs to choose a gauge. One common gauge is the ...
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Quantum momentum (De Broglie)
The de broglie hypothesis suggests a particle can be associated with a wave of
momentum $p = \hbar k$
my question is the following: how does one arrive at this concept of the momentum of a wave?
I ...
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Really why does promoting numerical variables to operators neatly work?
Apparently nice duality between classical and quantum mechanics first noticed by Dirac. As a graduate student of mathematics I believe such a wonderful similarity in their mathematics have a deep root ...
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Hydrogen energy levels and energy-time uncertainty principle
Some hydrogen atom exists in some excited quantum state, and after some time $\Delta t$ it's de-excited, emitting a photon carrying the energy difference.
It is claimed that this photon will carry ...
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Find the Bogoliubov transformation $b=SaS^\dagger$ induced by the squeezed operator
A definition a bogoliubov transformation is defined as $$b=ua+va^\dagger~,~ b^\dagger=u^*a^\dagger+v^*a$$
But, using squeeze operator $$S=\exp{\left[\frac{1}{2}(z (a^\dagger)^2-z^*a^2)\right]}$$ we ...
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Violation of Bell-like inequalities with spatial Boltzmann path ensemble: Ising model?
Quantum mechanics is equivalent with Feynman path ensemble, which after Wick rotation becomes Boltzmann path ensemble, which can be normalized into stochastic process as maximal entropy random walk (...
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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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Localization of Electron Matter Field Excitation in Simple Electron QFT Model
I believe QFT represents a single free (stationary) electron as a an excitation of the electron matter field which then couples to the EM field to create a local 'attached' EM field - if this is ...
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Why is $\textbf{S}_1\otimes \textbf{S}_2 = \sum_{i = x,y,z}S_{1i}\otimes S_{2i}$?
I'm looking at the spin-squared operator for a two-particle state, and I've been confusing myself about justifying the equality in the posting title. Consider the vector space $V\otimes W$, and two ...
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What is the preferred basis objection to the many-worlds interpretation of quantum mechanics?
I've seen the preferred basis problem referred to in many places, but have not seen a clear explanation of what the problem is. For example, this question asks whether the problem has been solved, but ...
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Lieb-Robinson Bound for bosonic systems
Background
Let us restrict our discussion to bosons and adopt the convention First Quantised $\leftrightarrow $ Second Quantised Theory (we are following these Ashok Sen's Quantum Field Theory I of ...
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Are identical particles always entangled even when not interacting?
Aren't the states of two identical particles always entangled even if they are not interacting? The states of two identical particles are either symmetric or antisymmetric i.e., cannot be written as ...
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Can eigenstates of a Hilbert space be thought of as delta functions?
Say we have an observable that describes a Hilbert space and that observable acts on state kets. Lets take the position observable for example. Then $\langle y|x\rangle = \delta(y - x)$. But can the ...
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On state transformations and the requirement of space-time invariance in (non-relativistic) quantum mechanics
I am trying to follow the development in Ballentine's Quantum Mechanics: A Modern Development but am struggling a lot. Please excuse my attaching of a picture of the development, but my question quite ...
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Quantum input-output theory : Why do we multiply by density of mode to have a number of photon **per unit of time**
In this paper, https://journals.aps.org/pra/abstract/10.1103/PhysRevA.31.3761, we work with input-output theory. I will first summarize the physics of it and then ask my question.
In input-output ...
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Commutation of Hamiltonian with momentum
In which case does the Hamiltonian $H$ commutes with the momentum $P$?
Can anybody help me? With an example? (No particular or strange Hamiltonians and no particular momenta are involved).
How can I ...
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Landau level degeneracy in symmetry gauge, finite system
As we know, Landau level degeneracy in a finite rectangular system is $\Phi/\Phi_0$, where $\Phi=BS$ is the total magnetic flux and $\Phi_0=h/q$ is the flux quanta. This can be easily derived using ...
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Generalized coherent states in arbitrary potential
It is well known that for Hamiltonian of harmonic oscillator there exist special states which saturate the Heisenberg uncertainity principle and under the time evolution follow closely classical ...
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Can quantum mechanics be formulated without any reference to pictures?
NOTE: in the following with the word "picture" I refer to Schroedinger, Heisenberg, Interaction pictures, i.e. to the way the time-evolution is "distributed" between states and operators.
We often ...
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How does an electron "move" in an $s$-orbital?
I have read multiple answers on StackExchange about this question, but I wasn't able to find a concrete answer. Like other questions, the reason I ask about the $s$-orbital is because it has a zero ...
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Nuclear Spin of Sodium 23
I am actually calculating the nuclear spin of Sodium 23. Here we have 11 protons and 12 neutrons. Now both the nuclei are short of the magic numbers. When I use the shell model for protons and ...
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Importance of complex functions in quantum mechanics
In quantum mechanics, we work with the space $\mathcal{H} = L^2(\mathbb{R})$ of functions with complex value square integrable. Thus Hermitian operators will play a central role since they have a real ...
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Expanding two-variable function $f(x,y)$ over the complete sets $\{ g_{i}(x) \}$ and $\{ h_{j}(y) \}$
Quite often (see, for example, this PDF, 50 KB) when discussing the Born-Oppenheimer approximation the following assertion is made: any well-behaved function of two independent variables $f(x,y)$ can ...
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Operational definition of rotation of particle
The question in brief: what does it mean, operationally, to rotate an electron?
Elaboration/background: I am trying to understand how representation theory applies to quantum mechanics. A stumbling ...
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State of a system in Quantum Mechanics and state vectors
I'm taking a course in Quantum Mechanics and there is something I'm not being able to fully understand. On more elementary courses on Quantum Mechanics I've been told that the idea of Quantum ...
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Normalisation of free particle wavefunction
The wavefunction $\Psi(x,t)$ for a free particle is given by
$$\Psi(x,t) = A e^{i(kx-\frac{\hbar k}{2m}t)}$$
This wavefunction is non-normalisable. Does this mean that free particles do not exist in ...
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Why do coherent states behave semi-classically, but harmonic oscillator states do not?
A coherent state of the quantum harmonic oscillator is defined as an eigenvector $|\alpha\rangle$ of the annihilation operator $\hat a$ with eigenvalue $\alpha$ or as spatial translations of the ...
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How projective representations can lead to 't Hooft anomalies in quantum mechanics?
In Shao's talk https://youtu.be/2vTvHYYl1Qk?t=1554, he argues that in quantum mechanics "if a symmetry acts projectively on states, then we have a t' Hooft anomaly". But I'm having trouble ...
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fixed input qubit state to an arbirary pure state using two variable rotations and one fixed rotation
It is a theorem that any arbitrary unitary transformation in SU(2) can be factored into the following form:
$ O = U_X(\theta) U_Y(\phi) U_X(\delta) $
Where $U_X$ is a Bloch sphere rotation. I ...
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How can any quantum superposition get produced, given conservation of energy?
Imagine I have a two-level quantum system $Q$ (motivating example: a trapped-ion qubit) where there is some energy difference between the states $\vert 0\rangle_Q$ and $\vert 1\rangle_Q$. Suppose I ...
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Is there an anticommutator relation for orbital angular momentum?
So I know that there are commutator relations for $L$ such as $[L_x,L_y] = i\hbar L_z$, but is there a relation for the anticommutator? For example, $L_xL_y + L_yL_x$?
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Momentum operator ambiguous?
In nonrelativistic quantum mechanics, are different operators possible as a candidate for the momentum operator, given that one has fixed one position operator and a hilbert space that this position ...
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Operators and periodic boundary conditions
Background:
In Ref. 1, a system of $N$ (identical) fermions is considered. The system is enclosed in a cubic box of volume $\Omega=L^3$ and periodic boundary conditions are employed, that is (I'll ...
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Distinguishability and tagging of particles
I was reading Sakurai's book and here is an extract:
In classical physics it is possible to keep track of individual particles even though
they may look alike. When we have particle 1 and particle 2 ...
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Different forms of Heisenberg Uncertainty Principle
The Heisenberg Uncertainty Principle is often written in two forms:
$$\Delta x \Delta p \geq \frac{\hbar}{2} $$
and
$$\sigma_x \sigma_p \geq \frac{\hbar}{2}. $$
Are these two equivalent? I've been ...
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How would the eigenstates of a particle with spin 3/2 look like?
I learnt in an introductory course about quantum mecanics how to work with spin 1/2 particles. I saw how the algebra is almost the same as for angular momentum, but no one ever told me about particles ...
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Different Hilbert spaces in quantum mechanics
In a lot of literature the notion of different Hilbert spaces has been mentioned. In QFT, for non-interacting theories the Hilbert space is a called a 'Fock space', this is different from the ...