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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451 views

Interpretation of boundary conditions in time-independent Schrödinger equation

The time-independent Schrödinger equation: $$\ -\frac{\hbar^2}{2m}\frac{d^2\psi}{dx^2} + V\psi = E\psi$$ is second order, so we should expect the solution to have two "degrees of freedom" which can ...
11
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2answers
528 views

Tensor product in quantum mechanics?

I often see many-body systems in QM represented in terms of a tensor products of the individual wave functions. Like, given two wave functions with basis vectors $|A\rangle$ and $|B\rangle$, belonging ...
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1answer
46 views

Allowed Wave Functions of System

Given a single-particle system with Hamiltonian $H$, what constraints can be put on the wave function at a particular point in time $\psi(x)$? Of course $\psi(x)$ must obey boundary conditions given ...
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0answers
44 views

How to find the minimum value of potential in QM?

In MIT problem sets I followed a solution of an exercise which focuses on odd-parity energy eigenstates in finite square well. The point of problem is how to know or find the minimal value of ...
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3answers
106 views

Same quantum states represented in different basis

In literature on an introduction to quantum mechanics which I am working through, there is a section which explains that a vector has different representations based on the basis you choose and then ...
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1answer
26 views

Reflection in Finite Square Wells

For a Finite Square Well where we have a wavefunction $\psi(x)$ which is an energy eigenfunction with eigenvalue $E = 2V_0$ in the following potential: $V(x) = \begin{array}{ll} 6V_0 ...
7
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191 views

What do the wave functions associated to the Fock states of each mode of a bound state system mean?

$\renewcommand{\ket}[1]{\left \lvert #1 \right \rangle}$ Consider a string of length $L$ under tension and clamped on each end. This system is described by the wave equation and has a set of modes. ...
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1answer
41 views

Wavefunction of a system of particles

A three-dimensional volume $V$ contains a certain number $N$ of electrons and they can't escape the volume $V$. Assume for simplicity that the potential $\mathcal{V}(\mathbf{r})$ is zero in all the ...
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1answer
134 views

What's the relation between molecular orbitals and electron density?

The way molecular orbitals are drawn represent the "encapsulated" space in which the wave function has a significant amplitude. How do I obtain from this the electron density? Is there a fundamental ...
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3answers
47 views

Position and momentum measurement effects on wave functions

I have a few short questions about an interpretation of what happens with position and momentum wave functions described in literature I am using. Given momentum space wave function and position space ...
2
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2answers
1k views

Solving quantum radial equation for infinite potential spherical annulus for $l=0$

There is a mass $m$ in a potential such that $$ V(r) = \left\{ \begin{array}{lr} 0, & a \leq r \leq b\\ \infty, & \text{everywhere else} \end{array} \right. $$ ...
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1answer
54 views

Can there be a wave function that is physically possible but is non differerentiable (maybe even non-continous)?

The definition of a wave function demands continuity and differentiability so that it can satisfy the Schrödinger Equation. My question is whether this assumption is necessary for reality. Does ...
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36 views

Singlet and triplet excited states in He atom

I found the following example for Term symbol usage in my coursebook: There are two electrons in He atom. Let the first one $e_{1}$ be in ground state, with $n_{1}=1$, $l_{1}=0$, $m_{l1}=0$, ...
0
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1answer
59 views

Would a collection of entangled particles behave like a superfluid? [closed]

Superfluidity of a Bose-Einstein condensate comes from the fact that all the particles are found in the same quantum state. They are described by the same macroscopic wavefunction. They never collide ...
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2answers
59 views

Observer in the double slit experiment with photons

In the double slit experiment with photons, the interacting observer is an instrument, detector… If you replace the detector with a piece of metal with the same mass as the mass of the detector, the ...
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1answer
40 views

Correlating two definitions of bound states in quantum mechanics

In Griffiths, he defines a bound state to be that stationary state for which the total energy E is such that $E<V(\pm\infty)$. Let $\psi(x)$ is a stationary state satisfying $E<V(\pm\infty)$ and ...
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0answers
50 views

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 ...
0
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1answer
46 views

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?
2
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1answer
82 views

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 ...
3
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2answers
46 views

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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1answer
22 views

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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1answer
260 views

Finite potential well, parity of solutions

I'm working through some problems for a QM exam and I've realised I don't really understand the concept of parity of solutions. I'm looking at a simple finite potential well problem: $$V(x)=0, \quad ...
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4answers
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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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42 views

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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19 views

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
259 views

How to visualize a Schrödinger cat state?

I recently read about Schrödinger cat states, which are basically a superposition of two coherent states $|\alpha\rangle$ with opposite phases, that is, $$ |\mathrm{cat}\rangle = |\alpha\rangle \pm ...
2
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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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47 views

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 ...
3
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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
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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3answers
94 views

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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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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2answers
22 views

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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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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5answers
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Is the wave function of a particle re-created after a measurement stops?

Yeah, I haven't quite understood, or been told, what happens to, for example an electron and its wavefunction, when you stop to measure it. I mean, an electron has a wave function describing its ...
3
votes
1answer
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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1answer
114 views

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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2answers
401 views

Finite Square Well Inside an Infinite Square Well

Ok here's a potential I invented and am trying to solve: $$ V(x) = \begin{cases} -V_0&0<x<b \\ 0&b<x<a \\ \infty&x>a \\ \end{cases}$$ and $V(-x) = V(x)$ (Even ...
15
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2answers
577 views

The formal solution of the Schrodinger equation

Consider the Schrödinger equation (or some equation in Schrödinger form) written down as $$ \tag 1 i \partial_{0} \Psi ~=~ \hat{H} \Psi . $$ Usually, one likes to write that it has a formal solution ...
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27 views

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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49 views

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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4answers
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Why do electrons in an atom occupy only the stationary states?

When we talk about the elementary problems in quantum mechanics like particle in a box, we first calculate the energy eigen-function. Then we say that the most general state is the linear combination ...
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0answers
41 views

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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2answers
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 ...
2
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2answers
88 views

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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0answers
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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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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 ...
2
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1answer
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 ...
3
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1answer
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 ...