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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Wave function of particle and antiparticle

The wave functions of particle and antiparticle are related by complex conjugation and wavefunction Ψ must be complex for particle such as n, p. Is there way to prove this mathematically? Can we do ...
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How to calculate probability of complex wave functions? [closed]

An election has an equation as such: $$Ψ(x) = e^{iαx^2}.$$ How am I supposed to find the probability of finding the electron over a certain range? Is Fourier Transform involved in this?
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Is there a way to prove that a bound state wavefunction can always be chosen real for an arbitrary potential in Quantum Mechanics?

As we can prove many things that always (at least in introductory quantum mechanical problems) apply using an arbitrary potential (like that $E>V_{\rm min}$ or else the solutions are ...
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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: ...
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Gauge transformation of wave function of a system of stationary charges

Let's say we have a system of $n$ stationary charges interacting via Coulomb potential. Let's ignore possible external electromagnetic fields. Moreover the system is quantum, and its wave function is ...
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Interpretation of a case of the double square well

Consider a 'double square well' potential (where $E < 0$) defined as: $$V(x) :=\begin{cases} -V_0~~~~~~~~~~~~~\text{for }\frac{b}{2} < x < \frac{b}{2} + a \text{ and } -a - \frac{b}{2} < ...
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wave propagation modelling

what is the best modelling technique for modelling mm-wave propagation in electromagnetic environment. Right now,am working on how to use use Transmission-line matrix (TLM) and ray-tracing techniques
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1answer
100 views

Particle here at a given time, in another galaxy a second later… Really?

I read "The Quantum Universe (Cox & Forshaw)" that a particle can be measured at a given position at a given time, and in another galaxy one second later. The probability of such event may be ...
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Three particles case, finding ground energy state

Here I came up with three particles in a box problem. (Assumption: Here I do not consider the interaction between particles and spin for simplicity.) What I want to do is express the ground state's ...
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1answer
32 views

Representing an ISW wavefunction graphically [closed]

I'm trying to decode this diagram given to us for an assignment. The description of the diagram is 'Consider a particle of mass m confined to a 1-dimensional square well, given graphically by the ...
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1answer
82 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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1answer
81 views

Calculating the energy of a particle using the Time Independent Schrodinger Equation [closed]

If we have a wave function $\Psi(x,t=0)$ which is a solution to the TISE for a zero potential in an infinite square well, would calculating the energy at $t = 0$ at a position be as easy as ...
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Exchange principle in terms of states and coordinates?

I have seen the exchange principle written in two ways, one in terms of coordinates and the other in terms of states: If $\psi_{AB}(1,2)$ represents particle $A$ in state $1$ and particle $B$ in ...
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Meaning of the Vector Wave Equation

So I thought I would try my luck here on physics stack exchange about an intuitive meaning of the Vector Wave Equation. I know there are a lot of resources out there that explain this equation, but ...
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36 views

How do we know that there is a wavefunction which collapse?

How do we know that there actually is a wavefunction in the first place which collapse. How do we know that there is a transition from some linear combination of the eigenfunctions to a single one? ...
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118 views

Significance of $i$ in the Schrödinger equation [duplicate]

There's an imaginary $i$ in the Schrödinger equation, which I guess is to define the position of the particle in a space-time involving a complex function. But what is the real physical significance ...
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1D transmission lines wave equation solution

you may know that the solution of 1D wave equation by d’Alembert is F(x-ct)+F(x+ct) and my question is that like is this F(x-ct) at transmission lines only the equation of one forward going wave that ...
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The Hilbert space that contains the first order correction to the state vector in Time-independent Perturbation Theory

When deriving the expression for the first order correction to the state vector of the new hamiltonian( H = H0 + H' ) we assume that $|\psi$n1> = $\sum_{m \neq n}$ C$_m$(n) $|\psi ^0 _m>$ ...
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Wave function for step potential

Given the step potential $$V(x)=\begin{cases} 0~~~~~~~~\text{if }~~x \leq 0 \\ V_0~~~~~~\text{if }~~x > 0 \end{cases}$$ Consider the case where $E < V_0$. In this region $x \leq 0$ we have ...
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Help normalising and taking the inverse Fourier transform of this wavefunction [closed]

Normalising Consider the wavefunction $$\psi(x,0)=Ne^{-\frac{|x|}{\lambda}}.$$ In order to normalise this I take the integral, which due to the modulus on the $x$ I evaluate just from zero to ...
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Connection between Hamiltonian version of the least action principle and probability amplitude in the Schrödinger equation

If I'm not mistaken, Schrödinger was influenced to look at wave equations because of de Broglie's assertion about particles having a wavelength. He started with the Hamiltonian equation which is ...
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67 views

Quantum Mechanics: How to compute how fast must a function go to zero at infinity? [closed]

We say that the wave function must go to zero at infinity faster than $1/x^{0.5}$ in order for it to be normalizable. What about other quantities like the probability current? What is the general rule ...
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Normalization of a wave function in quantum mechanics

A more simple question, so I am watching a quantum mechanics lecture on potentials of free particles and am doing the general solution of schrodinger's stationary equation for a free particle when I ...
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Expanding a wavefunction [closed]

I have a wave function that I have already normalised: $$ \psi(x) = \sqrt{\frac{30}{a^{5}}}x(a-x) $$ but now I have been asked to expand it to get: $$ \psi(x) = \sqrt{\frac{960}{\pi^{6}}}\sum_{k} ...
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136 views

About 'de Broglie hypothesis' and the double slit experiment

EDIT: As i mentioned in my original question, i do not have the background to fully understand @Timaeus answer (which was very detailed indeed). I would appreciate if someone could give a more ...
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3answers
193 views

On the completeness relation in Quantum Mechanics

Why does $$ \sum_n \Phi^{\ast}_n(x)\Phi_n(r)=\delta(x−r) $$ represents a completeness relation? Or, put differently, why does it imply completeness? Is there any way to see it intuitively? Maybe an ...
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55 views

Why is the reflection coefficient in quantum mechanical scattering defined this way?

In Griffiths' "Introduction to Quantum Mechanics, second edition" section 2.5.2, p. 73, he states: For the delta-function potential, when considering the scattered states (with $E > 0$), we have ...
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56 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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27 views

How to tell when exponentials are real valued? (Barrier Potential)

Where $k_1=\frac{\sqrt{2mE}}{\hbar}$ and $\alpha=\frac{\sqrt{2m(V_0-E)}}{\hbar}$ I'm quite confused as to why the exponentials in regions I and III are complex functions while in region II the ...
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1answer
36 views

Existence of representation of symmetry transformation

There is a simple fact that we can change our point of view and that physical laws should remain the same, id est, outcomes of our experiments should be the same no matter from which frame of ...
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Perodic boundary conditions vs Dirichlet?

I have been working through several examples recently involving particles in boxes (when finding the partition function of an ideal gas for example or looking at photon gases). I have seen two ...
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43 views

Aharonov-Bohm effect [closed]

In the build up to Aharonov-Bohm effect, one has to represent the gauge covariant form of STIE. We need to consider two things; the vector potential A changes`under gauge transformation and as a ...
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Difference between a wavevector and wavefunction

I often see both terms used in textbooks, but I am not sure whether I understand the difference between them. Both describe the state of a system, however, they seem different in some ways. From what ...
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1answer
59 views

General expectation value

I have a basic question related to finding expectation values of an operator $\hat{Q}$ We know that the expectation of $\hat{Q}$ (in the position space) is given by $$\langle Q \rangle=\int {\Psi^* ...
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107 views

Derivation of Schrodinger's wave equation

To derive $$i \hbar \frac{\partial}{\partial t} \psi = H \psi,$$ we start with $$i \hbar \frac{\partial}{\partial t} |\alpha \rangle = H| \alpha \rangle$$ and then multiply by $\langle x|$ on the ...
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61 views

Perturbation by electrical field in infinite potential well: difference in first energy corrections because of difference in the limits of the well

In time independent perturbation theory we can calculate the first and second energy corrections resulted by a potential V in the Hamiltonian $ H=H_o + λV , $ , λ<<, by the expressions: $$ε_1 = ...
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1answer
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Probability of finding electron in a spherical shell

In the book Arthur Beiser - Concepts of modern physics, probability of finding an electron in Hydrogen atom in the spherical shell between $r$ and $r+dr$ is given as \begin{equation} P(r)dr = ...
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Continuous spectra and quantum decoherence

Suppose that some quantum wave function $\psi = \int a_i \,i\rangle \,\,di$ where pseudo-spanning ket "vectors" of $\psi$, $i \rangle$, are continuous. (thus the use of integral.) By normalization, ...
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Understanding wave functions of matter waves

The wave functions of matter waves give the probability density of the particle being at a certain location. Does this arise because as an outside observer, we have incomplete information about the ...
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88 views

Momentum and position for free particle

In the section of 'The free particle' in 'Introduction to quantum mechanics, second edition' by Griffiths page 65. He has the wave equation as $$\Psi(x,t) = \frac{1}{\sqrt{2 \pi}} ...
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Bloch Functions as an implication of the Crystallographic Restriction Theorem?

I'm studying Bloch Functions and it seems to me safe to assume that they are the most general Eigenfunction of a Hamiltionian with the crystal periodicity. Now the only considerations made in deriving ...
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The time evolution in Dirac delta potential [closed]

We know that the dirac delta potential has exactly one bound state. If the potential strength suddenly changes a value, the bound state should evolve to the new bound state, how to describe the time ...
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Where the time-dependent wavefunction $\Psi(\vec{x},t)$ lies?

Supose $\vec{x}=(x,y,z)\in \mathbb{R}^3$. The state of a physical system is described by the function $\Psi(\vec{x},t)$, where it must satisfy $$\int_{\mathbb{R}^3} ...
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64 views

Is the sum of two stationary states of different energies also a stationary state?

The question title kind of speaks for itself really. I was thinking of maybe using the orthogonality relation to try to show this: $$\int_{-\infty}^{\infty}\phi_n(x)\phi_m(x)dx=\delta_{nm}.$$ ...
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Does an expectation value over only part of a wavefunction have physical meaning?

Does the expression: $$\langle p\rangle_{x=a, b} =\frac{\int_a^b \Psi(x)^*\,\hat{p}\,\Psi(x)dx}{\int_a^b |\Psi(x)|^2dx} $$ have any physical meaning when $\int_a^b ...
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How can a probability distribution have wavelength (de Broglie wavelength)?

The wave function described by Schrodinger's equation is interpreted as describing the probability of a particle in at any point in space, i.e. a probability distribution. Since this distribution ...
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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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Help needed to understand “On the reality of the quantum state”

I am having trouble to understand the reasoning in the following paper, On the reality of the quantum state. MF Pusey, J Barret and T Rudolph. Nature Phys. 8, 475–478 (2012); arXiv:1111.3328. ...
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Deriving Wave Function for Scattering States with Delta-Function Potential

I am following the Griffiths Book on Quantum Mechanics, and am following the derivation for the wave function for Delta-Function Potentials. $$V(x) = -\alpha \delta(x)$$ In the scattering states, ...
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How can I solve this quantum mechanical “paradox”?

Let a (free) particle move in $[0,a]$ with cyclic boundary condition $\psi(0)=\psi(a)$. The solution of the Schrödinger-equation can be put in the form of a plane wave. In this state the standard ...