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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135 views

Is there a reason why probability density is written as $\psi^*\psi$ instead of $\psi\psi^*$?

As the title states, I see $|\psi|^2$ written as $\psi^*\psi$ instead of $\psi\psi^*$. Are both correct or is there a reason behind it? As far as I'm aware, the only time I see this sort of ordering ...
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1answer
212 views

Analogy to Fourier transform in spherical coordinates with boundary at a certain radius

Suppose, we have a wavefuction $\phi(\vec{x})$ which is restricted in a sphere, with the spherical boundary condtion $$\phi(\vec{x}=R)=\phi_0.$$ How can I do the 'Fourier transformation' as the case ...
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1answer
84 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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1answer
86 views

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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0answers
83 views

Guess the wave function in a given potential

Are there any techniques in guessing the ground state wave function in any given potential? For example, for a given potential like $$ \frac{1}{1-x^2}$$ or $$ \frac{1}{1-x^3}~?$$ I know wave ...
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2answers
84 views

Measurement of energy apparently violating the position-momentum Uncertainty Principle in a potential that does not depend on distance?

I am taking a beginning course in QM and I have learnt that the measurement of energy collapses the wavefunction of a particle to one of its energy eigenstates. But some misconceptions regarding this ...
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2answers
96 views

Normalisation of free particle wavefunction

The wavefunction $\Psi(x,t)$ for a free particle is given by $$\Psi(x,t) = A e^{i(kx-\frac{\hbar k}{2m}t)}$$ This wavefunction is non-normalisable. Does this mean that free particles do not exist in ...
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0answers
44 views

Additional quantum states of the infinite square well

The quantum states $\psi(x)$ of the infinite square well of width $a$ are given by $$\psi(x) = \sqrt{\frac{2}{a}}\sin\Big(\frac{n \pi x}{a}\Big),\ n= 1,2,3, \dots$$ Now, I understand $n \neq 0$, as ...
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3answers
99 views

Modern explanation of the Young experiment with Quantum Field Theory?

In the Young double slit experiment it is possible to detect the arrival of individual photons as well as an interference pattern. It doesn't makes much sense to me that something could be either a ...
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1answer
92 views

Bound states of Dirac Delta function in infinite well

If there is a potential of $-\alpha\delta(x)$ for $-a<x<a$ and $\infty$ elsewhere, and the energy of the system is less than 0, then I'm trying to find the wave function. From the Schrodinger ...
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0answers
32 views

How come an electron's wave function being nonzero at far distances doesn't mean it can travel faster than light? [duplicate]

I think the wave function of a free electron is nonzero almost everywhere. In particular there are regions of space arbitrarily far away where the electron has positive probability of being found. If ...
2
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1answer
74 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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5answers
357 views

Derivation of Schrödinger equation - free particle

I learn quantum physics from Alonso-Finn's book (Amazon link), there's one step of Schrödinger equation for a free particle that I couldn't understand. $$ \frac{\mathrm{d^{2}\Psi } }{\mathrm{d} ...
2
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1answer
46 views

Antisymmetry requirement for the total wavefunction

My understanding is that if we are dealing with a system of two electrons, the total wavefunction needs to be antisymmetric only when the two electrons have same value of n and l ( i.e. they are ...
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2answers
727 views

What is the Copenhagen interpretation of quantum field theory?

I am most interested in interpretational differences due to the fact that quantum field theory is relativistic while quantum mechanics is not. By "Copenhagen interpretation" I mean a minimal ...
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2answers
49 views

Measurement of position after collapse of a wavefunction

Suppose I have a wavefunction which collapses to a certain eigenstate after a measurement of energy. In that state, I perform a calculation of position and obtain a certain position value, say $x_0$. ...
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4answers
783 views

How does wave function collapse when I measure position?

Text books say that when you measure a particle's position, its wave function collapses to one eigenstate, which is a delta function at that location. I'm confused here. A measurement always have ...
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0answers
43 views

Showing two wavefunctions are proportional to one another [duplicate]

I am struggling to answer the following question: Let ψ₁(x) and ψ₂(x) be normalisable energy eigenfunctions for a particle of mass m in one dimension moving in a potential V(x). Suppose that ψ₁ and ...
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2answers
2k views

Matter waves and de Broglie wave length

The wavelength of a particle of momentum p is calculated using De Broglie relation. The de Broglie relation was postulated for what is called a matter waves. Now according to the statistical ...
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2answers
112 views

How do we decide whether an electron orbital has a non-zero or zero probability of lying inside the nucleus of an hydrogen atom?

How do we decide whether an electron orbital has a non-zero or zero probability of lying inside the nucleus of an hydrogen atom? It is mostly from the radial function, as to what I think but how ...
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1answer
105 views

Schrodinger equation violates mathematics?

By the Hamiltonian formalism of quantum mechanics, given a quantum system in a state $\Psi$ in a Hilbert space $\mathcal H$, the state will instantaneously evolve in time according to ...
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0answers
34 views

Is the wavefunction of particles inside a gas spread or localized?

For an individual free particle that starts localized, the wave function packet spreads over time, so the particle becomes less localized. Suppose now that we have a gas of those particles inside a ...
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1answer
162 views

Continuity behaviour of a wavefunction when the potential exhibits a discontinuity

If a potential $V(x,t)$ exhibits a finite discontinuity in space, the wavefunction $\phi(x,t)$ and its spatial derivative will be continuous. If a potential exhibits a finite discontinuity in time, ...
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2answers
188 views

Time dependent and time independent Schrödinger equations

I'm trying to understand the relation between the time dependent and time dependent Schrödinger equations. In particular, we know that the TDSE is $$H\Psi=i\hbar \frac{\partial \Psi}{\partial t}$$ ...
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0answers
52 views

solutions of wave equation with cubic term

Does the following equation $$ \nabla^\mu \nabla_\mu \psi + a \psi^3 = b \psi $$ where $\psi$ is a real function, $a$ and $b$ are real constants, have other solutions that extend beyond a one ...
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1answer
92 views

Infinite potential well question with wave function $\psi (x) = (x -a/2)^2$

In an infinite potential well with width $a$, a particle in this potential well is at state with wave function is $\psi (x) = (x -a/2)^2$ (not normalized). If you measure the energy of ...
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3answers
91 views

Quick way to compute $\langle n^{'}l^{'}m^{'}|r^k|nlm \rangle$, $k \in I$; $|nlm\rangle$ is $H$ atom eigenfunction [closed]

I want to compute quickly (using maybe some scaling arguments) $\langle n^{'}l^{'}m^{'}|r^k|nlm\rangle$, where $k \in I$. $|nlm \rangle$ is the eigenfunction of the Hydrogen atom ($H$). Example: ...
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1answer
77 views

Confused on how to interpret the energy eigenfunction of Hydrogen

So here is an image of the third lowest energy eigenfunction of an electron in a hydrogen atom: Image from http://imgur.com/Lu4MocL I understand well the eigenfunctions given by Schrodinger's ...
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1answer
42 views

Spinor expectation value and measurement

I have a question about the difference between expectation value and probability of measurement. consider the spinor $\zeta = [-3\ \ 4i\ ]^T$ . The expectation value of $S_x$ is zero because : ...
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2answers
101 views

Coefficients and wavefunction in quantum mechanics

In general quantum mechanics we represent the state of a system with a state vector $| \psi \rangle $ in some Hilbert space in some base. Assuming a complete discrete set of bases vectors $ |n \rangle ...
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2answers
223 views

Born Oppenheimer Approximation: Why can any molecular state be represented as a linear combination of electronic states?

in the Born Oppenheimer Approximation, one expands the molecular wavefunction $\Psi(x,X)$ in terms of the electronic wavefunctions $\phi(x;X)$: $$\Psi(x,X)= \sum_k(c(X)_k\phi(x;X)_k)$$ ($x$ are the ...
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1answer
203 views

Calculating the Reflection Coefficient of a Potential Step Explicitly

So I'm using the following definition for the Reflection coefficient, $R$ : $$R=\frac{\left\rvert\ \vec{j}_{reflected}\right\rvert}{\left\rvert\ \vec{j}_{incident}\right\rvert}$$ where ...
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2answers
130 views

Mathematical confusion in quantum mechanics

During a class about Ehrenfest theorem, my teacher use an equation to proceed its derivation (to prove $\frac{d<r>}{dt}=\frac{<p>}{m}$ ) and that is: ...
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1answer
85 views

A wave function that is normalized initially remains normalized

Suppose that $\Psi(x,t)$ is normalized at time $t=0$. Show that this implies that $\Psi(x,t)$ is normalized at all other times. I know that this makes intuitive sense, and we'd certainly want our ...
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1answer
118 views

Is a non-degenerate wavefunction real or complex?

In this video it is stated that: It can easily be verified that the wavefunction of a non-degenerate quantum mechanical system will be real. However the presenter does not explain why this ...
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3answers
635 views

Can we measure “wavefunction” of quantum particles?

We know that there is uncertainty principle, so question: can we ever measure wavefunction of particles? I do not think this is possible, but I am not sure. I guess that everything is probabilistic. ...
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0answers
71 views

Why must the probability be the integrated square modulus of the wave function [duplicate]

Quantum mechanics uses the wave function to calculate probabilities by taking the square modulus of the wave function as requirement by Max Born. Why should this (squaring of the wave function) be so, ...
3
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2answers
145 views

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} ...
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1answer
50 views

Scattering off of a bi-local potential

I am trying to figure out the scattering wave function for the following potential: $$V(x,x')=-A \phi(x)\phi^*(x')$$ Such that the SE can be written as $$[\frac{\hbar^2\partial^2_x}{2m}-E]\psi = ...
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2answers
1k views

Linear vs. quadratic dispersion relation

In wave mechanics the dispersion relation between frequency $\omega$ and wave number $k$ is linear: $$\omega_n=c k_n$$ But in quantum mechanics, based on Schrödinger's equation, one can show that we ...
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0answers
58 views

Valence bands question

I'm currently doing solid state physics and learning about semiconductors. During the course, I have seen a lot of energy/wavevector graphs, like this one (pic from Kittel): I did not have a ...
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4answers
368 views

Does $\lvert\langle p\lvert\psi\rangle\rvert^2$ have any meaning at all?

I used to think $\lvert\langle p\lvert\psi\rangle\rvert^2$ had the meaning of some likelihood of the particle's momentum being $p$ (within some tolerance interval $\Delta p$). Now I'm just confused. ...
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0answers
59 views

Does wave function of an electron itself move?

As far as I know quantum mechanics, electrons in an atom in vacuum move accordingly a wave function (a complex scalar field), but the wave function itself does not move (except that the atom may ...
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1answer
63 views

Are Neutrons and anti-Neutrons attracted to each other over distance?

Lets create a scenario where you have a total vacuum and you're shooting into this vacuum two streams, one, a Neutron stream and the other an anti-Neutron stream and because you're curious what will ...
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3answers
89 views

Particle in a box: value for wave function $u(x)$ when potential $V(x)$ is infinity

The time-independent Schrödinger equation (TISE) is: $$ -\frac{\hbar^2}{2m}\frac{d^2 u(x)}{dx^2}+V(x)u(x)=Eu(x) \hspace{15pt}$$ where $E$ is a constant. Imagine now a infinity potential well as ...
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2answers
60 views

Angular momentum for a given Wavefunction [closed]

Problem: Give a particle in the state $\Psi = e^{\frac{-(x^{2}+y^{2}+z^{2})}{a^{2}}}(\frac{x}{a} + \frac{yz}{a^{2}})$, what are the allowed values for $l_{x}$ (and later for $l_{y}, l_{z}$). Attempt: ...
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1answer
47 views

What is the main difference between a free particle on a line and a free particle on a circle?

The energy spectrum for a free particle in a circle with radius $r$ is $$E_n=\frac{n^2\hbar^2}{2mr^2}.$$ The energy spectrum for a free particle on an infinite line is similar. If so, what is the ...
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1answer
109 views

Why are eigenspaces of a Hermitian operator mutually orthogonal? [closed]

In Quantum Mechanics, from the properties of the solution of Schrodinger's Equation inside the infinite well, is that they are: Mutually orthogonal for different eigenvalues. Orthonormal. Complete. ...
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1answer
75 views

The boundary condition for delta function

Beginning with the Schr\"odinger equation for $N$ particles in one dimension interacting via a $\delta$-function potential $$(-\sum_{1}^{N}\frac{\partial^2}{\partial ...
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3answers
421 views

How axiomatic is the symmetrization requirement (i.e. the Pauli principle)? (in QM)

I've so far always been told, that the symmetrization requirement is an axiom on the level of the Schrödinger equation and the statistical interpretation of the wave function (or it's absolute value). ...