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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Tensor product of two different Pauli matrices $\sigma_2\otimes\eta_1 $
I'm solving problem 3.D in H. Georgi Lie Algebra etc for fun where one is to compute the matrix elements of the direct product $\sigma_2\otimes\eta_1$ where $[\sigma_2]_{ij}\text{ and }[\eta_1]_{xy}$ ...
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What conservation law corresponds to this local $U(1)$ symmetry of the CCR?
It is known that canonical commutation relations do not fix the form of momentum operator. That means that if canonical commutation relations (CCR) are given by
$$[\hat{x}^i,\hat{p}_j]~=~i\hbar~\...
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Question on Sakurai's treatment of the Harmonic Oscillator:
In Section 2.3 of the second edition of Modern Quantum Mechanics (which discusses the harmonic oscillator), Sakurai derives the relation $$Na\left|n\right> = (n-1)a\left|n\right>,$$ and states ...
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Why did Dirac say that atomic time is different from relativistic time, and that gravity is becoming weaker? What is the relation between the two?
In this gem of an interview in 1982 with Friedrich Hund, Dirac says at 09:17 that there is some theoretical basis and observational evidence that atomic time and distances are different from ...
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Can average momentum be imaginary?
I am new to quantum physics. We just learnt about wave equations, observables and expectation values today. What really caught my attention was the expectation value of average momentum and energy:
$$\...
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Why is the ground state energy of a linearly perturbed quantum oscillator always lower than its harmonic counterpart?
I am concerned with a QHO that is linearly perturbed in $x$, i.e.
$$ H = \hbar \omega \left(\hat{n} + \frac{1}{2}\right) + \lambda \underbrace{\left(\hat{b}+ \hat{b}^\dagger \right)}_{\propto \hat{x}}....
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Is it theoretically possible for a person to pass through a solid wall/object?
I understand that matter cannot pass through other solid matter because of the electrons that orbit an atom prevents this but I was curious to know if it is theoretically possible to somehow get ...
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Flow of time in Quantum Mechanics vs General Relativity
I was reading a Wikipedia article about the Problem of time, which states:
quantum mechanics regards the flow of time as universal and absolute, whereas general relativity regards the flow of time as ...
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What is the difference between a photon and a phonon?
More specifically, how does a wave-particle duality differ from a quasiparticle/collective excitation?
What makes a photon a gauge boson and a phonon a Nambu–Goldstone boson?
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Can one define wavefunction for Bogoliubov quasiparticle excitation in a superconductor?
Wavefunction is essentially a single particle concept. It is easily extended to multiparticle system as follows- if one has say five electrons the wavefunction of this five electron state is any ...
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A particle in a 1D box: what is the meaning of velocity?
In the box $x = 0$ to $x = L$, $V = 0$, and for $x < 0$ and $x > L$, $V = \infty$ (infinite potential well).
The eigenvalues of the Hamiltonian are:
$$E_n = \frac{n^2 h^2}{8L^2} \, .$$
Since ...
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A two-level system absorbs a detuned photon. Where does the extra energy go?
Let's consider simple two-level system with frequency gap of $\omega_0$ between ground and excited state. Now, when we turn on external electromagnetic field with frequency $\omega < \omega_0$, ...
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What's the highest-$n$ Rydberg state that's been created and detected experimentally?
Rydberg states form an infinite series of electronic states that asymptotically approach the ionization potential of the atom or molecule, usually in good agreement with the simple Rydberg formula.
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Isn't a single Quantum one single string? [duplicate]
In physics, a quantum (plural: quanta) is the minimum amount of any physical entity involved in an interaction.
In Quantum Mechanics There is no difference between one Quantum to another one.
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What does it really mean for a state to be in a superposition?
Since a quantum information lecture today I have been wondering what does it really mean for a state to be in superposition? Is this something that is answerable?
This is what we learnt (or what I ...
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Dirac notation - specific acting orientation for operators
I have this doubt:
Imagine two operators $A$ and $B$ and the state $\psi$.
I know that the following statement is true:
$$\langle\psi| A|\psi\rangle^*=\langle\psi| A^\dagger|\psi\rangle$$
But is ...
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Quantum Collapse
When we observe a quantum object does it collapse into a point? Or does it collapse into a smaller wave of area that is out of our range of accuracy? My gut tells me the latter.
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Path integral kernel dimensions and normalizing factor
I am currently reading Quantum Mechanics and Path Integrals by Feynman and Hibbs. Working on problem 3.1 made me wonder why the 1D free particle kernel: $$ K_0(b,a) = \sqrt\frac{m}{2\pi i \hbar(t_a - ...
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Time ordering operator if commutator is $c$-number function
I have a question concerning the time ordering operator. Let's suppose we have a time evolution generated by some Hamiltonian $H(t)$ given by
$$
U(t)=T_\leftarrow\exp\left(-\mathrm{i}\int_0^t\mathrm{d}...
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Zero Point Fluctuations
The total energy of a mode in a quantum mechanical resonator is given by
$E_n ~=~ (n+ 1/2)hf$ where $n$ is the number of modes.
So when there are no modes or vibrations, i.e. $n=0$, the energy is ...
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How to solve double delta potential bound states by "brute force"
I just solved a problem in Griffiths' Intro to QM, where one had to find the bound states given the potential:
$$V(x)=-\alpha [\delta (x-a)+\delta(x+a)]$$
In order to solve it, one had to exploit the ...
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Time dependent Schrödinger equation with time independent potential and separation of variables
suppose we have a potential that's independent of time $V(x,t) = V(x)$
so in Schrödinger equation we get:
$$i\hbar \frac{\partial \Psi (x,t)}{\partial t}=-\frac{\hbar^2}{2m}\frac{\partial^2 \Psi (x,t)}...
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Adding Angular Momenta Operators in QM
Consider $j,m$ to be the angular momentum magnitude and $z$-projection eigenvalues corresponding to a total angular momentum operator $\hat{J}$, composed of angular momentum $\hat{J}_1$ and $\hat{J}_2$...
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If orbital shells are just probability functions, why are quantum numbers only ever integers? [closed]
Quantum numbers are supposed to denote every individual orbital. But if orbital shells are probability functions, then orbitals can't be definite, solid things. So in that case, there can be variation ...
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No well-defined frequency for a wave packet?
There are similar questions to mine on this site, but not quite what I am asking (I think). The de Broglie relations for energy and momentum
$$ \lambda = \frac{h}{p},
\\
\nu = E/h .$$
equate a ...
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Are combined fermion wavefunctions still antisymmetric after wavefunction collapse?
If we have two electrons in a state $|\psi\rangle=\frac{1}{\sqrt2}[|\uparrow\downarrow\rangle+|\downarrow\uparrow\rangle]$ and we measure the spin of the first electron to be up, does the wavefunction ...
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is decoherence continuous?
Pardon my naivete here. In a quantum system, it seems that even a few photons from the environment can decohere the entangled particles in the system in a trillion trillionth of a second ( or faster).
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From particles to fields - how to reconcile these two approaches?
I was watching some lectures of the theoretical minimum program, by prof. Leonard Susskind. There he introduces the notion of fields (in the context of QFT) in a very nice and intuitive way.
Suppose ...
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Quantum expressions involving Dirac delta function
I want to find the following quantum expression:
$$ \langle x|PX|x'\rangle.$$
A. If I use $X|x'\rangle = x'|x'\rangle $, I will get:
$$ \langle x|PX|x'\rangle = \langle x|Px'|x'\rangle = x'\langle x|...
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Electrons moving faster than light and backward in time?
In Lawrence Krauss's book "A Universe From Nothing"; page 62 mentions that for a very short period of time, so small it cannot be measured, an electron due to the uncertainty principle can appear to ...
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Whence the $i$ in QM Poisson bracket definition?
On p. 87 of Dirac's Quantum Mechanics he introduces the quantum analog of the classical Poisson bracket$^1$
$$ [u,v]~=~\sum_r \left( \frac{\partial u}{\partial q_r}\frac{\partial u}{\partial p_r}- \...
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Linearizing Quantum Operators
I was reading an article on harmonic generation and came across the following way of decomposing the photon field operator.
$$ \hat{A}={\langle}\hat{A}{\rangle}I+ \Delta\hat{a}$$
The right hand side ...
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What does the "quantum mutual information" quantify?
I'm having some difficulty understanding the physical meaning of the mutual information that two subsystems share with each other. For example, if $\rho_{AB}$ defines the matrix of a bipartite state, ...
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Where is quantization used in deriving Planck's law?
There are several explanations for how Planck used quantization to explain blackbody radiation correctly without the ultraviolet catastrophe. I will follow this explanation.
For a cavity, the mode ...
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What is the logic behind box normalization and periodic boundary condition?
Free particle energy eigenfunctions are $A\exp{[i(Et-\textbf{p}\cdot\textbf{r})/\hbar]}$ are non-normalizable. To normalize them one introduces a procedure called 'box normalization' where one imposes ...
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A moment of cohomology$.$
As is well-known (cf. Ref.1), the momentum operator is defined up to a time-independent closed form. More precisely, the physically inequivalent momentum operators are classified by the de Rham ...
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Can the pilot wave theory explain why the circumference of an orbit has to be an integer multiple of the wavelength of the electron orbiting it?
Consider the atomic model proposed by Bohr. The velocity of an electron at any orbit is given by the following equation:
$$v= n \dfrac{h}{2\pi mr}$$
Now, this equation stems from the fact that, quote, ...
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Very simple example of the way the Fourier transform is used in quantum mechanics?
According to a book I'm reading, the Fourier transform is widely used in quantum mechanics (QM). That came as a huge surprise to me. (Unfortunately, the book doesn't go on to give any simple examples ...
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Hamiltionian spectrum in unstable systems
I have heard that the eigenvalue of Hamiltonian in an unstable system can contain an imaginary part corresponding the tunneling. Is that true? If it is the case, then I am very confused about it.
Let ...
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Do eigenvectors of quantum operators span the whole Hilbert Space?
I am trying to solve an exercise in Shankar's QM book (concretely 4.2.1), and I am asked the probability of each possible value for the operator $L_x$ when the particle is in a certain eigenstate of ...
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How can we prove that a photon is absorbed only once?
When I first heard about the photons and the double-slit experiment my immediate thought was the following: Alright, energy is not absorbed continuously but in discrete units, photons, but nature ...
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What is a Hilbert Space?
The Hilbert Space is the space where wavefunction live. But how would I describe it in words? Would it be something like:
The infinite dimensional vector space consisting of all functions of ...
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Dirac-delta-functions as eigenbasis of the position operator - pure nonsense? Or can more be said?
I remember overthinking equations like
\begin{equation}
\mathbf{1}=\int dx\ |x\rangle\langle x|\tag{1}
\end{equation}
and
\begin{equation}
X=\int dx\ |x\rangle\langle x|x\tag{2}
\end{equation}
when I ...
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Trace in non-orthogonal basis?
Physicists define the trace of an operator $\rho$ as the follows,
$Tr(\rho)=\sum\limits_{|s\rangle \in B} \langle s| \rho |s\rangle$
where $B$ is some orthonormal basis, and this quantity is basis ...
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Higher orders in perturbation theory
I would like to compute an energy level up to many orders in perturbation theory. My difficulty right now is not in the calculation itself but in understanding the algebraic structure of the higher ...
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Help understanding proof in simultaneous diagonalization
The proof is from Principles of Quantum Mechanics by Shankar. The theorem is:
If $\Omega$ and $\Lambda$ are two commuting Hermitian operators, there exists (at least) a basis of common eigenvectors ...
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Entangled particles
So we have two particles (A and B) that are entangled.
From what I understand, entanglement isn't destroyed, it is only obscured by subsequent interactions with the environment.
Particle A goes ...
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Computer simulation of Schrödinger equation [duplicate]
I am looking for a computer program which simulates the Schrödinger equation (say for a single particle) in two dimensions and for potentials and initial states specified by the user.
Typical ...
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Is quantum mechanics truly probabilistic?
Probability arises inherently from a lack of information. For example, if I were to take a ball out of a bag with 3 yellow and 2 white balls, I would have a 0.6 probability of getting a yellow and a 0....
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Why is the Lagrangian approach preferred over the Hamiltonian approach in QFT? [duplicate]
Going from non-relativistic quantum mechanics(QM) to QFT there is a marked change in the approach used. QM almost exclusively uses Hamiltonains. Lagrangian based methods like the path-integrals are ...
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