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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Density of classical states in quantum theory

Let's first treat electrons as classical objects. I can evaluate the classical energy of each state in a configurational space (3N real numbers and, say, spins) using just Coulomb's law. Then I ...
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Lower limit for atomic number in case of imaginary ion with Z < 1

In my programming project I calculate the minimal energy of an atom with 2 electrons in the $L=0, S=0$ state, using a Hylleraas wave function. The values I find for $Z=2$ (He) and $Z=1$ (H$^-$) are ...
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31 views

What processes create or destroy information?

From a classical standpoint, it seems pretty clear that information can be easily lost. If you knock over a bookshelf and the books fall out, it seems like their initial order on the shelf cannot be ...
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132 views

How can energy be negative in a finite square well?

Say if the potential $V(x) < 0$ in the well but the sides or the scattered states its zero potential, anyways How is that the energy in the well is less than zero? Is it because the potential ...
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16 views

Solving traveling wave using the shooting method

The spatially-dependent Hodgkin-Huxley equation for a cylindrical dendrite or unmyelinated axon: where $\frac{a}{2\rho}\frac{\partial^2V}{\partial x^2}$ is a diffusion term $a$ is the fiber radius, ...
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14 views

Connection between singlet, triplet two-electron states and the Slater determinant

I'm confused about a number of things concerning two-electron systems and spin. Here is (perhaps too much) exposition, skip to "the problem" if you want: Consider the helium atom in the simplified ...
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48 views

How did Max Born come up with his rule? [duplicate]

In his rule, he stated that the probability is norm-squared of wave function, $|\psi|^2$. And as far as I knew, no one else at that time had "right" interpretation of the wave function. Even ...
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37 views

Particle in a Spherically Symmetric Box [on hold]

Problem The spherically symmetric potential energy of a particle with mass $m$ is given by $V(r)=0 $ if $ a<r<b$ $V(r)=\infty$ elsewhere where $r^2 = x^2+y^2+z^2$. (a) ...
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1answer
44 views

Can particle quantum spin be described with a wave function? [on hold]

I'm a little confused about the idea of spin. It's been non-technically described to me as "like magnetic dipole moment", except only two possible "directions". But I feel like that's a bad analogy, ...
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2answers
81 views

Is the Wave Function a Unitary Operator? [closed]

A unitary operator can be represented as an exponential $$e^{iA}$$ and as we represent the wave function in general as $$e^{i k x}.$$ Does that mean that the wavefunction is unitary as the exponent is ...
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25 views

Parts of the Quark Wavefunction

Quarks are fermions meaning that they have an antisymmetric wavefunction. Under particle exchange the sign of the wavefunction. The wavefunction is made up of a few different parts $$ \psi_{Total} = ...
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2answers
104 views

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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53 views

Confusion of Schrödinger equation and complex conjugates

I have a similar question that was asked in the following link: (Schrödinger's Equation and its complex conjugate). But I find both the question and answers not specific enough. So let me ...
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2answers
56 views

Linear Combinations of Energy Eigenfunctions in 1D

Given that a particle is in a state defined by the wavefunction: $$\Psi (x,t) = \psi_0(x)e^{-iE_0t/\hbar}+\psi_1(x)e^{-iE_1t/\hbar}$$ where $\psi_0(x)$ and $\psi_1(x)$ are the energy eigenfunctions of ...
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1answer
42 views

Wavefunction Collapse

I believe my Lecturer and the textbook have contradicted one another. My lecturer gave the example that if the spatial part of the wavefunction of a particle is given by $\psi(x) = c_1\psi_1(x) + ...
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Solving the Schrodinger Equation

What is meant by solving the stationary schrodinger equation? I have a well of width $L$ where $v(x)=-v$ inside the well and 0 outside I am trying to solve the schrodinger equation but i dont ...
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1answer
37 views

1D Scattering Phase Shift (Finite Well) - Unphysical?

I am calculating the phase shift from a 1-dimensional potential well. This seems extremely simple, but I am just getting so confused by it. Let there be a potential well of depth $V_0$ and spatial ...
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What is a wave function in simple language?

In my textbook it is given that 'The wave function describes the position and state of the electron and its square gives the probability density of electrons.' Can someone give me a very ...
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1answer
27 views

Taking Measurements of Quantities in QM

I have a quick question relating to Annihilation and Creation operators, and in taking observables in general. Let's say, for instance, that I prepare a particle so that I consider the projection of ...
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2answers
53 views

What do operations on single Qubits of Unfactorable Superpositions Do?

So suppose I have the following Quantum Circuit: A ---- |Control| -----|Hadamard|---- B ---- |xxxxxxx|------------------------ Which is a 2 input Controlled Gate (applying some gate of two choices ...
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2answers
2k views

Schrödinger's Equation and its complex conjugate

I would like to know why there is a minus sign on the right-hand side of the Schrödinger's complex conjugate equation, whereas in the Schrödinger's equation there isn't. I know it is a simple ...
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32 views

Are $\psi ^{*}(x,t)$ and $\psi(x,-t)$ solutions of the same Schroedinger equation?

I have this question: Let $\psi(x,t)$ solution of the Schroedinger equation for a particle under a potential V(x) independent of time. Are $\psi ^{*}(x,t)$ and $\psi(x,-t)$ solutions of the same ...
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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 ...
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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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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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43 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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102 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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502 views

Quantum Mechanics in Electric Field

I am working on a problem which looks like this. Consider a charged particle with charge $q$ trapped in a box of length $L$ with finite constant potential $ V_0 $ on both ends. A constant (static) ...
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1answer
25 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 ...
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181 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
40 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
128 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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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 ...
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1answer
172 views

Probability and double slit

if a beam of identical particles at random distances from each other (or exactly 1/2 lambda between each other) travelling with the same v towards a double sllit do not interfere with each others wave ...
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2answers
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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
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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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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 $n_{1}=1$, $l_{1}=0$, $m_{l1}=0$, ...
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1answer
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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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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
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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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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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1answer
43 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?
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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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1answer
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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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1answer
237 views

Semi-infinite / Asymmetric potential well

I'm asked to come up with an ansatz and solve for the coefficients of a asymmetric infinite potential well, where: $$ V = \begin{cases} \infty \text{ for } x< 0 \\ V_0 \text{ for } 0 \leq x \leq L ...
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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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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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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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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} < ...