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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Solution of 1-D Schrodinger equation for the potential $V(x) = -\frac{1}{|x|}$

May be this question might have already been asked but i couldn't find it, so let me know if its already there. Consider a potential, $V(x) = -\frac{1}{|x|}$ and if we apply this to a one dimensional ...
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639 views

Solving a time independent Schrödinger equation with a given potential

I'm trying to rework this old homework problem, and I am having problems arriving at the same solution on the answer sheet: Let $$V(x)=\begin{cases}\infty &\text{ if } x < 0\\ ...
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147 views

Wavefunction operators and the observable [closed]

So I got this from the exam I had yesterday. I couldn't really answer it other and it played on my mind through the night Show that if a wave function $\psi$ , is an eigenfunction of an operator [Q], ...
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Writing wave functions with spin of a system of particles

Suppose I have 2 fermions in a potential $V(x)$. Both particles are moving in one dimension: the $x$ axis. Then, neglecting the interaction between the particles, the spatial wave function of the ...
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250 views

A quantum particle which is almost at rest but whose position is random!

Assume a particle is given by a quantum state which is constructed in such a way that it is equally probable to find it anywhere in an fixed interval $(0,L)$ but has arbitrarily low velocity. The ...
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361 views

Schrödinger equation in position representation

$$ \DeclareMathOperator{\dif}{d \!} \newcommand{\ramuno}{\mathrm{i}} \newcommand{\exponent}{\mathrm{e}} \newcommand{\ket}[1]{|{#1}\rangle} \newcommand{\bra}[1]{\langle{#1}|} ...
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No well-defined frequency for a wave packet?

There are similar questions to mine on this site, but not quite what I am asking (I think). The de Broglie relations for energy and momentum $$ \lambda = \frac{h}{p}, \\ \nu = E/h .$$ equate a ...
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Decomposition of this wave function in eigenfunctions

I have this wave function of a system on a central potential: $V(r)$: $$\Phi(x,y,z)=C(x+y+z)e^{-\alpha r^2}.$$ And I'm asked a few things about probabilities. I don't have problems with that, because ...
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247 views

Problem from Sakurai about a delta-function potential [closed]

Can you help me with this problem from Sakurai: A particle of mass m in one dimension is bound to a fixed center by an attractive delta-function potential: $$V(x) ~= ~-a\delta(x) , \qquad ...
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791 views

Expectation value of momentum

I'm having a problem with an expectation value that doesn't seem to add up for me. What I know is, that $\psi(\vec{r})$ is a wavefunction for a particle in three dimensions. The Hamiltonian is given ...
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Text interpretation in Griffith's intro to QM

It says in Griffith's chapter 2.1, that: $$\tag{2.14} \Psi(x,t)~=~\sum_{n=1}^{\infty}c_n\,\psi_n(x) e^{(-iE_n t/\hbar)}$$ It so happens that every solution to the (time-dependent) Schrodinger ...
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Expectation value of position in infinite square well

I'm looking for some help to a question. I'm working in the infinite square well, and I have the wavefunction: $$\psi(x,t=0)=A\left( i\sqrt{2}\phi_{1}+\sqrt{3}\phi_{2} \right).$$ For every time t, ...
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55 views

First-order pertubation theory

I'm having some trouble figuring this out, so I was hoping someone could help. I need to show that the first-order pertubation of the ground state energy is not changed by the pertubation $H'$, given ...
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271 views

Infinite Potential Well Energy for Piece-wise Constant Wave Function

I'm trying to compute the expectation value of energy for a certain state in an infinite potential well but I'm getting contradictory answers. The well has potential \begin{align} V(x) = \left\{ ...
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Physical interpretation of normalization of wave fuctions

Does normalization of wave function mean that we are getting our state vector to unit length? If that's the case what does it mean physically? Also is the underlying vector space finite dimensional? ...
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803 views

Momentum of particle in a box

Take a unit box, the energy eigenfunctions are $\sin(n\pi x)$ (ignoring normalization constant) inside the box and 0 outside. I have read that there is no momentum operator for a particle in a box, ...
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191 views

Harmonic oscillator - wavefunctions

I understand now how I can derive the lowest energy state $W_0 = \tfrac{1}{2}\hbar \omega$ of the quantum harmonic oscillator (HO) using the ladder operators. What is the easiest way to now derive ...
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Meaning of inner product $\langle \vec{r} | \psi(t)\rangle $

I have come across the equation which comes out of the nothing in Zettili's book Quantum mechanics concepts and applications p. 167: $$\psi(\vec{r},t) ~=~ \langle \vec{r} \,|\, \psi(t) \rangle.$$ ...
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114 views

What is the wave length of the entire universe?

In quantum physics, particles are also waves. Larger particles have shorter wave lengths, and macroscopic objects have extremely short wave lengths so that the wave aspect can be ignored, and ...
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298 views

How can particles travel in a straight line?

A particle can be set off in a certain direction by giving them momentum. Momentum is a vector, so the particle heads off in a specific direction. But the wave function of the particle allows it to ...
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465 views

Using the Normalization Condition with Wavefunction

I'm very confused with this problem and I was looking for some guidance. $$\psi(x) = Ae^{ikx}e^{-x^2/2a^2}$$ Use the normalization condition to find A. So I understand that you use the normalization ...
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What does the wavevector $\textbf{k}$ mean?

In Ashcroft, Mermin Solid State Physics, Eq. 17.43 is $$ \epsilon(\textbf{k}) = \frac{\hbar^2 k^2}{2m} - e\phi(\textbf{r}) $$ where $\textbf{k}$ is the wavevector and all other symbols have their ...
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Interpretation of de Broglie wave

Until what point can the de Broglie wave be thought as a real wave? I mean, is it made of something? What amplitude does it have? Is it a sine wave? How can it be related to the wavefunction of the ...
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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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Periodic boundary condition on a Wave Function of a Particle in a Box

Until now solving the Schrodinger Equation for a particle in a box was relatively easy because the boundaries conditions imposed zero value on the wave function at the boundaries. But now I must find ...
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Time evolution of Gaussian wave packet

I'm slightly confused as to answer this question, someone please help: Consider a free particle in one dimension, described by the initial wave function $$\psi(x,0) = ...
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236 views

Electron in an infinite potential well

Does this problem have any sense? Suppose an electron in an infinite well of length $0.5nm$. The state of the system is the superposition of the ground state and the first excited state. Find the ...
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425 views

Bohr-Sommerfeld quantization from the WKB approximation

How can one prove the Bohr-Sommerfeld quantization formula $$ \oint p~dq ~=~2\pi n \hbar $$ from the WKB ansatz solution $$\Psi(x)~=~e^{iS(x)/ \hbar}$$ for the Schroedinger equation? With $S$ the ...
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A general wavefunction in a square lattice

Suppose we have a square lattice with periodic condition in both $x$ and $y$ direction with four atoms per unit cell, the configuration of the four atoms has $C_4$ symmetry. What will be a general ...
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102 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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647 views

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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209 views

Time Dependent HydroHow would I go about writing the time dependent wave function given the wavefunction at $t=0$? gen Wave Function

1) How vwoulHow would I go about writing the time dependent wave function given the wavefunction at $t=0$? go about writing the time dependent wave function given the wavefunction at $t=0$? ...
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332 views

Mathematical explanation of quantum teleportation

I am now studying quantum teleportation. I get what the process is like but I'm wondering why it happens this way. You've got two entangled particles A and B whose wavefunctions are entangled. You ...
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169 views

Nodes and Antinodes for standing wave

In the arrangement shown in the figure below, an object of mass m can be hung from a string (linear mass density $\mu$ = 2.00 g/m) that passes over a light (massless) pulley. The string is connected ...
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Vector representation of wavefunction in quantum mechanics?

I am new to quantum mechanics, and I just studied some parts of "wave mechanics" version of quantum mechanics. But I heard that wavefunction can be represented as vector in Hilbert space. In my eye, ...
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Why does a plane wave have definite momentum?

Apologies if this is a little vague. It might not have a good answer. Given the interpretation of $|\psi(x)|^2$ as a probability distribution it's unsurprising that a wave function that is ...
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376 views

Potential step and its transmission / reflection

Lets say we have a potential step with regions 1 with zero potential $W_p\!=\!0$ (this is a free particle) and region 2 with potential $W_p$. Wave functions in this case are: \begin{align} ...
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How does one find the wave velocity and the phase speed?

While I was studying beats, I tried to find a displacement function of any particle in the most generalized form. I ended up with $$y=2A\sin(\pi(t-x/v)(f_1+f_2))\cos(\pi(t-x/v)(f_1-f_2)).$$ Now, ...
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262 views

Why the hydrogen radial wave function is real?

Why the hydrogen radial wave function is real? Is it a coincidence?
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Why do people say the phase oscillates in time and the amplitude stays the same but the intensity of a traveling beam does oscillate with time?

I'm confused why people say the phase oscillates in time and the amplitude stays the same (the reason for having complex numbers). But on the other hand, the intensity of a traveling beam does ...
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Why does the wave description say that probability oscillates, while the phase interpretation says constant amplitude?

The wave description of a particle illustrates an oscillating probability of the particle being found in any point in space. When a particle travels, it carries along with it a phase that oscillates ...
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160 views

normalizing a wavefunction

I have a homework problem that I can't get started on, below is the first bit. I feel like I should just be able to integrate to find $C$ but I get a divergent integral. Can someone give me a hint as ...
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Is there anything to prevent paired-up neutrons from a complete overlap

The reason "neutrons don't overlap", as DarenW explained it, has to do with intricate forces at play that take into account the spins, iso-spins and symmetry of the wavefunctions. However, assume I ...
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74 views

Why is the Horizontal Force Constant in Deriving the One Dimensional Wave Equation

My textbook in deriving the wave equation for a one dimensional elastic string stated that the horizontal direction force is constant.I understand that the horizontal components of the tensions on ...
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635 views

Finite, square, potential well

Lets say we have a finite square well symetric around $y$ axis (picture below). I know how and why general solutions to the second order ODE (stationary Schrödinger equation) are as follows for ...
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Is normalization consistent with Schrodinger's Equation?

Schrodinger's Equation does not set a limit on the size of wave functions but to normalize a wave function a limit must be set. How is this consistent physically and mathematically with Schrodinger's ...
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Phase shift in electromagnetic potential

In Aharonov-Bohm effect, how to derive that the wave function of a electric charge $q$ acquires a phase shift $\phi=\frac{q}{\hbar}\int \mathbf{A} \cdot d\mathbf{x}$ after travelling in the non-zero ...
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Question about the linearity of wave functions

For piece-wise constant potential, the potential energy is constant so the time dependent wave function can take the form $\psi(x,t)=C_1e^{i(kx- \omega t)}+C_2e^{i(-kx-\omega t)}$ where ...
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133 views

Where is a particle bound in a delta potential?

I can picture a bound state in a harmonic oscillator, or in an infinite square well, but where is a particle bound in a delta potential?
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Wavefunction restrictions of odd potentials

So I was just reading back through Griffiths' "Introduction to Quantum Mechanics" and solving some of the problems for practice. There is a nice one (problem 2.1c for those playing at home) where you ...