A complex scalar field that describes a quantum mechanical system. The square of the modulus of the wave function gives the probability of the system to be found in a particular state.

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Where does $\hat{P}\psi(x) = -i\hbar \partial_x \psi(x)$ come from?

It's a very basic question, where does the relation $$\hat{P}\psi(x) = -i\hbar \partial_x \psi(x)$$ for any square integrable $\psi(x)$ come into existence? Some texts I found states that the above ...
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Do wave functions really belong to $L^2$ space, or do we need to restrict our physical Hilbert space even further?

I am beginning to study quantum mechanics and I got stuck right at the beginning. I am trying to prove that the time derivative of the expected value of momentum of a particle is the (negative) ...
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88 views

What happens to the wave function of a particle immediately after measuring its energy?

For this question, I will be adhering to the Copenhagen interpretation (since that's what I've learned in university so far). For the sake of brevity/clarity, also, assume the Hamiltonian here has ...
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680 views

Physical reason why the derivative of a wavefunction has to be continuous?

Question What is the physical reason (i.e. without any maths) that the derivative of a wavefunction (except with infinite potentials) has to be continuous? Other info I know that in the classical ...
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186 views

How do you determine the “phase” of a hydrogen eigenfunction?

I've been reading the wikipedia article on the atomic orbitals of hydrogen. They have a nice collection of diagrams, such as this one for n,l,m = 3,1,1 This is apparently showing the wavefunction, ...
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92 views

Quantum Wavefunctions Without Space

A handful of physicists have a rather peculiar definition of 'nothing' in terms of cosmology. Their claim is that the Universe, assuming it has 0 total energy, could have arisen from nothing but ...
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308 views

Why is $\omega = \sqrt{K/m}$ valid for a quantum oscillator?

I'm working in the 3rd edition of Modern Physics by Serway, Moses, and Moyer. In 6.6, it talks about a quantum oscillator. I don't fully understand how the definition of frequency works. Now, we ...
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274 views

Logic of the 'imaginary wave function collapse' argument in Double Slit experiment

My question is in regards to the stance that the 'wave function collapse' is not an actual physical occurrence. That is, you are not, by observation, changing the particles position from a wave to a ...
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159 views

Why do we use $\psi$ instead of a straightforward probability?

What is the advantage/purpose of using $\psi$ for wavefunctions and getting the probability with $|\psi|^2$ as opposed to just defining and using the probability function?
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135 views

At what point is the spin determined in a Stern-Gerlach Apparatus

Consider a particle with spin that travels through a Stern Gerlach box (SGB), which projects the particle’s spin onto one of the eigenstates in the $z$-direction. The SGB defines separate trajectories ...
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How to find the wavefunction that solves an infinite square well with a delta function well in the middle?

Solutions for the wavefunction in an infinite square well with a delta function barrier in the middle are easily found online (see here for an example). I am wondering what the wavefunction is for an ...
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233 views

Workaround to fermion sign problem?

My (rather incomplete) understanding of the NP-hard fermion/numerical sign problem is that it occurs when attempting to converge on a wavefunction for many-body fermion systems (for example, a small ...
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Wavefunction as a combination of two stationary states - how to find those states?

Lets say we have a particle in a infinite square well which has a wavefunction like this ($A$ is some constant and $d$ is the width of the well): \begin{align} A\left[ \sin \left(\frac{2 \pi x}{d}\...
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732 views

wavefunction collapse and uncertainty principle

We all know that wavefunction collapse when it is observed. Uncertainty principle states that $\sigma_x \sigma_p \geq \frac {\hbar}{2}$. When wavefunction collapse, doesn't $\sigma_x$ become $0$?, as ...
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41 views

Protocol for solving time independent Schrodinger equation

Just a short question about the protocol for solving the time-independent Schrodinger equation for different potentials and the reasons for accepting and rejecting solutions. Take for example the ...
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61 views

Should state vectors be considered constant?

By the principle of superposition, a state vector can be defined as $$\begin{align} \psi(x) &= c_1 \psi_1(x) + c_2 \psi_2(x) + \cdots + c_n \psi_n(x) \\ \lvert\psi\rangle &= \begin{pmatrix}c_1 ...
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90 views

How do we find the number of bounded states in this potential?

for the potential $$V(x)=-\frac{1}{1+\frac{x^2}{m^2}}$$ we can approximate the wave function and bounded state accurately for $x << m$ as simple harmonic oscillator, so what are we gonna do if ...
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Exact closed form solution to the quantum harmonic oscillator

I came across this question in Griffiths QM, which asked to show that this equation $$\Psi(x,t)=\left(\frac{m\omega}{\pi \hbar}\right)^{1/4} \exp\left[-\frac{m\omega}{2\hbar} \left(x^2+\frac{a^2}{2}(...
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222 views

Plane wave expansion of cylindrical functions:Summation of the Hankel functions

I understand that; in cylindrical coordinates, the basic solutions of the Helmholtz equation are of the form Hankel function of integer order times a complex exponential term ($E=H_{n}(kr)e^{in\alpha}...
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298 views

Infinitely many degeneracy of Landau level: Countable or Uncountable?

Description of Landau levels can be found in many standard textbooks of quantum mechanics and here. Two ubiquitous solutions apply either the symmetric gauge $\vec{A}=(-\frac{1}{2}By,\frac{1}{2}Bx,0)$ ...
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305 views

Probability of measuring momentum [closed]

Suppose we have this wavefunction: $$ \psi = A \left( cos(kx) + cos (2kx) \right) $$ I have to find the possible results of measurement of momentum and their probabilities. Attempt For a momentum ...
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347 views

Harmonic Oscillator potential, proof that Gaussians remain Gaussians?

I read in several papers that for a Harmonic Oscillator Hamiltonian in the time dependent Schrödinger equation a Gaussian wave packet remains Gaussian. Unfortunately I could not find any proof for ...
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118 views

Connection between a simple matter wave and Heisenberg's uncertainty relation

When looking at the wave function of a particle, I usually prefer to write $$ \Psi(x,t) = A \exp(i(kx - \omega t)) $$ since it reminds me of classical waves for which I have an intuition ($k$ ...
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Hydrogen wave function in momentum space

We can seperate the wave function of an hydrogen atom in a radial and an angle part: $$ \phi_{n,l,m} (\mathbf{r}) = R_{n,l,m}(r) Y_{l,m}(\vartheta,\varphi) \, , $$ where $Y_{l,m}$ are the spherical ...
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Does quantum mechanics predict instantaneous action at a distance even without entanglement?

The suggestion that quantum mechanics implies that instantaneous action at a distance occurs is normally based on the contention that this follows from the entanglement of particles that share a ...
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Finding $\psi(x,t)$ for a free particle starting from a Gaussian wave profile $\psi(x)$

Consider a free-particle with a Gaussian wavefunction, $$\psi(x)~=~\left(\frac{a}{\pi}\right)^{1/4}e^{-\frac12a x^2},$$ find $\psi(x,t)$. The wavefunction is already normalized, so the next thing to ...
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Does a wave function describe the motion of electrons or atoms?

I took the course of quantum mechanics a while ago. I do not quite remember all the detail on how to derive the wave function for hydrogen but I still remember the general picture. I think the text ...
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Behavior of atom's wave packets in a gas

It is my understanding that the wave packet of a free localized particle spreads with time. My question is what is the best description of the particles in a gas inside a closed container: Do they ...
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86 views

What is a linear polarized photon?

According to Dirac a 'linear' polarized photon is a superposition of left and right rotating photons. Here is a puzzling aspect of this superposition. There are dichroic materials which can absorb ...
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Classical Limit of the Quantum Harmonic Oscillator

The classical harmonic oscillator obeys an arcsine law in that the distribution of positions of the particle over a single time cycle is proportional to $\frac{1}{\sqrt{A^2-x^2}}$, $A$ being the ...
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338 views

1D Finite potential well: solutions with $\sinh$ and $\cosh$?

So I am studying the (one dimensional) quantum mechanical finite potential well defined by: $$ V(x) = \cases{0, &|x|>a\cr -V_0, &|x|<a} $$ where $V_0>0$ is a real number. I know ...
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Koopmann von Neumann (KvN) Theory

I was just wondering does anyone have any informative sources apart from the obvious wikipedia articles regarding Koopmann von Neumann (KvN) theory? Or if its possible could someone explain the basic ...
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Normalising a wave function in parts?

If we have the wave function $\psi_{100}(r,\theta,\phi)=R_{10}(r)Y_{00}(\theta,\phi)$ when we are normalising it we do the following: $$1=\int| \psi_{100}(r,\theta,\phi)|^2sin(\theta) r^2drd\theta d\...
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428 views

Double slit experiment and entanglement [duplicate]

Just wondering, what would happen in this experiment. In the experiment you would first have two entangled particles. Then you fire one of the particles, lets say "Particle A", at a double slit ...
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394 views

Formulation and probability of a wave-function [closed]

I have got this problem where I have been given the following wave function: $$\Psi = 0\quad\text{if}~|x| > a\quad\text{and}\quad A(a^2-x^2)\quad \text{if} \quad |x|< a$$ Now the first question ...
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Implicit Postulate of Quantum Mechanics

Consider the following quantum system: a particle in a one dimensional box (= infinite potential well). The energy eigenstates wave functions all vanish outside the box. But the position eigenstates ...
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711 views

Expectation values-Wavefunction

I'm a bit puzzled about an excercise in which I have to find the expectation values for position and momentum. Normally this should be pretty easy but in this case I just don't get the point. ...
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111 views

What methods exist for us to measure the position and momentum of atoms that make up molecules?

In this paper, Localization of an atom by homodyne measurement. A. M. Herkommer et al. Quantum Semiclass. Opt. 8 no. 1, p. 189 (1996) (paywalled). the authors are able to localize atoms using ...
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Interpretation of $e|\psi|^2$ as electron density

In solid state physics the electron density is often equated to $e|\psi|^2$. However, the Sakurai says (Chapter 2.4, Interpretation of the Wave Function, p. 101) that adopting such a view leads "to ...
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Non-Degeneracy of Eigenvalues of Number Operator for Simple Harmonic Oscillator [duplicate]

Possible Duplicate: Proof that the One-Dimensional Simple Harmonic Oscillator is Non-Degenerate? I'm trying to convince myself that the eigenvalues $n$ of the number operator $N=a^{\dagger}a$ ...
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397 views

Why Pauli exclusion instead of electrons canceling out?

To quote Wikipedia, The Pauli exclusion principle is the quantum mechanical principle that no two identical fermions (particles with half-integer spin) may occupy the same quantum state ...
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696 views

Transmission and reflection

What is the transmission amplitude of a wavefunction $\phi(x)=e^{ikx}(\tanh x -ik)$? I would have thought that it is $\tanh x -ik$ since this is the factor associated with the forward travelling $e^{...
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2answers
240 views

Why does the wavefunction have to be continuous in the presence of a Dirac delta potential?

Considering the time-independent Schrödinger equation, I can see for a finite potential, why the wavefunction has to be continuous, I can also see why the first derivative of the wavefunction is ...
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1answer
43 views

What is the equation of motion for multiple simultaneous pressure waves in a medium? (In the context of stimulated Brillouin scattering)

My overall motivation is to derive the behavior of Brillouin scattering in a birefringent fiber. Brillouin scattering is a nonlinear interaction between light and sound. In classic Brillouin ...
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What is the connection between gravitons and geometry?

I know there are two ways to do quantum gravity. One can pick a background space-time (usually Minkowski flat space-time) and then at any time slice one can define the state of the universe as the ...
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How do you determine the degree of localization of a wavefunction?

Suppose that there is a wavefunction $\Psi (x,0)$ where 0 is referring to $t$. Let us also say that $a(k) = \frac{C\alpha}{\sqrt{\pi}}\exp(-\alpha^2k^2)$ is the spectral contents (spectral amplitudes) ...
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Quantum Mechanics: Can the probability of finding a particle in the whole space be smaller or higher at certain times?

In the book Introduction to Quantum Mechanics (by David Griffith) there is an Example 2.1: Suppose a particle starts out in a linear combination of just two stationary states: $$\Psi(x,0)~=~c_1\...
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Loss of interference in single-photon Mach–Zehnder interferometer with detector in only one arm

I have read that if you have a Mach–Zehnder interferometer (doing a single-photon experiment) and put a non-destructive detector in only one of the two arms (connected to the first beam splitter), you ...
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In quantum descriptions of atoms why are observables (which we derive from the wave function) attributed to electrons?

For example the orbital angular momentum, for the hydrogen atom. Is this the total angular momentum of the atom(electron and proton) or just the electron? I am asking because, I am learning about how ...
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Relation between p+ip wave Superconductor and Moore-Read State

I am quite interested in the understanding of the relation between p_ip wave superconductor(SC) and the Moore-Read(MR) state. They share many similar properties, for example, p+ip SC has majorana as ...