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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Why is $\omega = \sqrt{K/m}$ valid for a quantum oscillator?

I'm working in the 3rd edition of Modern Physics by Serway, Moses, and Moyer. In 6.6, it talks about a quantum oscillator. I don't fully understand how the definition of frequency works. Now, we ...
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113 views

Why do we use $\psi$ instead of a straightforward probability?

What is the advantage/purpose of using $\psi$ for wavefunctions and getting the probability with $|\psi|^2$ as opposed to just defining and using the probability function?
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91 views

At what point is the spin determined in a Stern-Gerlach Apparatus

Consider a particle with spin that travels through a Stern Gerlach box (SGB), which projects the particle’s spin onto one of the eigenstates in the $z$-direction. The SGB defines separate trajectories ...
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211 views

Why does $\ell=0$ correspond to spherically symmetric solutions for the spherical harmonics?

In quantum mechanics why do states with $\ell=0$ in the Hydrogen atom correspond to spherically symmetric spherical harmonics?
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885 views

Wavefunction as a combination of two stationary states - how to find those states?

Lets say we have a particle in a infinite square well which has a wavefunction like this ($A$ is some constant and $d$ is the width of the well): \begin{align} A\left[ \sin \left(\frac{2 \pi ...
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959 views

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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564 views

Can we impose a boundary condition on the derivative of the wavefunction through the physical assumptions?

Consider the Schrödinger equation for a particle in one dimension, where we have at least one boundary in the system (say the boundary is at $x=0$ and we are solving for $x>0$). Sometimes we want ...
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Smoothness constraint of wave function

Is there anything in the physics that enforces the wave function to be $C^2$? Are weak solutions to the Schroedinger equation physical? I am reading the beginning chapters of Griffiths and he doesn't ...
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163 views

Infinitely many degeneracy of Landau level: Countable or Uncountable?

Description of Landau levels can be found in many standard textbooks of quantum mechanics and here. Two ubiquitous solutions apply either the symmetric gauge $\vec{A}=(-\frac{1}{2}By,\frac{1}{2}Bx,0)$ ...
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65 views

Normalising a wave function in parts?

If we have the wave function $\psi_{100}(r,\theta,\phi)=R_{10}(r)Y_{00}(\theta,\phi)$ when we are normalising it we do the following: $$1=\int| \psi_{100}(r,\theta,\phi)|^2sin(\theta) r^2drd\theta ...
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61 views

Probability of measuring momentum [closed]

Suppose we have this wavefunction: $$ \psi = A \left( cos(kx) + cos (2kx) \right) $$ I have to find the possible results of measurement of momentum and their probabilities. Attempt For a momentum ...
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203 views

Harmonic Oscillator potential, proof that Gaussians remain Gaussians?

I read in several papers that for a Harmonic Oscillator Hamiltonian in the time dependent Schrödinger equation a Gaussian wave packet remains Gaussian. Unfortunately I could not find any proof for ...
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105 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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904 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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576 views

Even and Odd States of a 1D finite potential well

Is it possible for a particle trapped in a 1D finite potential well to evolve from a even state to an odd state and vice-versa? Why?
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506 views

wavefunction collapse and uncertainty principle

We all know that wavefunction collapse when it is observed. Uncertainty principle states that $\sigma_x \sigma_p \geq \frac {\hbar}{2}$. When wavefunction collapse, doesn't $\sigma_x$ become $0$?, as ...
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114 views

1D Finite potential well: solutions with $\sinh$ and $\cosh$?

So I am studying the (one dimensional) quantum mechanical finite potential well defined by: $$ V(x) = \cases{0, &|x|>a\cr -V_0, &|x|<a} $$ where $V_0>0$ is a real number. I know ...
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299 views

Formulation and probability of a wave-function [closed]

I have got this problem where I have been given the following wave function: $$\Psi = 0\quad\text{if}~|x| > a\quad\text{and}\quad A(a^2-x^2)\quad \text{if} \quad |x|< a$$ Now the first question ...
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488 views

Expectation values-Wavefunction

I'm a bit puzzled about an excercise in which I have to find the expectation values for position and momentum. Normally this should be pretty easy but in this case I just don't get the point. ...
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85 views

What methods exist for us to measure the position and momentum of atoms that make up molecules?

In this paper, Localization of an atom by homodyne measurement. A. M. Herkommer et al. Quantum Semiclass. Opt. 8 no. 1, p. 189 (1996) (paywalled). the authors are able to localize atoms using ...
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245 views

Interpretation of $e|\psi|^2$ as electron density

In solid state physics the electron density is often equated to $e|\psi|^2$. However, the Sakurai says (Chapter 2.4, Interpretation of the Wave Function, p. 101) that adopting such a view leads "to ...
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Finding $\psi(x,t)$ for a free particle starting from a Gaussian wave profile $\psi(x)$

Consider a free-particle with a Gaussian wavefunction, $$\psi(x)~=~\left(\frac{a}{\pi}\right)^{1/4}e^{-\frac12a x^2},$$ find $\psi(x,t)$. The wavefunction is already normalized, so the next thing to ...
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408 views

Is the electron wave function defined during photon emission

I have heard the term quantum leap to describe the (instantaneous?) transition from a higher energy orbital to a lower energy orbital. Yet, I understand that this transition time has now been ...
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574 views

Transmission and reflection

What is the transmission amplitude of a wavefunction $\phi(x)=e^{ikx}(\tanh x -ik)$? I would have thought that it is $\tanh x -ik$ since this is the factor associated with the forward travelling ...
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224 views

Majorana wavefunction

I'm trying to compute the wavefunction for a Majorana state in an nanowire/superconductor hybrid system, like arXiv: Majorana Fermions and a Topological Phase Transition in ...
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101 views

Functionals of quantum states in QFT

Almost every book and article I can think of represents states of QFT using the Heisenberg picture of Hilbert space vectors, but Visser in "Lorentzian wormholes" does mention that you can also ...
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How do I figure out the probability of finding a particle between two barriers?

Given a delta function $\alpha\delta(x+a)$ and an infinite energy potential barrier at $[0,\infty)$, calculate the scattered state, calculate the probability of reflection as a function of ...
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660 views

Is there any operator behind probability, in quantum mechanics?

In Quantum mechanics, the probability of finding a particle at position $x$ is given by $|\psi(x)|^2$, where $\psi$ is the wave function. Wonder what is the operator which gives this probability? Is ...
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Why do wave packets spread out over time?

Why do wave functions spread out over time? Where in the math does quantum mechanics state this? As far as I've seen, the waves are not required to spread, and what does this mean if they do?
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230 views

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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12k views

Sinusoidal Wave Displacement Function

I am learning about waves (intro course) and as I was studying Wave Functions, I got a little confused. The book claims that the wave function of a sinusoidal wave moving in the $+x$ direction is ...
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290 views

How do I integrate $\frac{1}{\Psi}\frac{\partial \Psi}{\partial x} = Cx$

How do I integrate the following? $$\frac{1}{\Psi}\frac{\partial \Psi}{\partial x} = Cx$$ where $C$ is a constant. I'm supposed to get a Gaussian function out of the above by integrating but don't ...
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415 views

What is the rationale behind representing a state function by a complex valued function in QM?

What is the rationale behind representing a state function of an electron with a complex valued function $\Psi$. If only the probabilistic argument was required then why not represent it with just a ...
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263 views

States and observables in quantum mechanics

I'm beginning learn quantum mechanics. As I understand, state is a map $\phi$ from $L^2(\mathbb R)$ such that $|\phi|^2$ describes probability density of a particle's position. By integrating ...
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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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225 views

How to understand wavefunction in quantum mechanics in math

I am reading some introduction on quantum mechanics. I don't understand all but I get the point that the wavefunction tells some probability aspects. In one book, they show one example of the ...
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329 views

wave superposition of electrons and quarks

Is quantum wave superposition of electrons and quarks possible? If not, can different types of elementary particles be mixed in wave superposition?
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Probability current

Conservation of probability: Suppose a wavefunction has ${\partial \mathbb P \over \partial t} = -t f(x,t)$ and ${\partial j \over \partial x} = i f(x,t)$. How does it follow that ${\partial \mathbb P ...
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165 views

Normalizing wavefunction

If you are trying to normalize $\psi = A\sin kx$, and you find that $|A|^2 = \frac{2}{a}$, why do you take the positive square root and not the negative? What happens if you take the negative square ...
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423 views

Double slit experiment and single particles. Is the wave function just a mathematical model?

I really do want to apologize in advance, I know this question has been 'answered' before. I have this 'problem' of feeling negatively toward much of today's 'mystical' interpretations of physics, ...
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161 views

Is $v(p)\exp(ipx)$ really the positron wave function?

In many textbooks the negative energy solution of the Dirac equation is quoted as describing the positron. Actually I don't understand this. For me $v(p)\exp(ipx)$ is the wave function of an electron ...
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121 views

Confused over the presence of 2 expressions for $\Psi(x,t)$

I'm following Griffiths' Introduction to Quantum Mechanics, and I see that he's got 2 different expressions for $\Psi(x,t)$. One of them is ...
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77 views

Minimum information necessary to represent a pure quantum state

I was thinking about how quantum states are represented for various types of systems, and how the amount of classical information (bits) required to represent a state depends on its basis. Let's take ...
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347 views

Wave function not normalizable

Does the solution of the Schrodinger equation always have to be normalizable? By normalizable I mean, given a wavefunction $\psi(x)$ $$\int_{-\infty}^{\infty}|\psi(x)|^2 dx<\infty \qquad ...
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153 views

Does Schrödinger equation have dual-property with Heat equation?

I have experimental data that Schödinger equation maintains high frequencies, while heat equation low. Does Schrödinger equation have some duality property with heat equation?
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216 views

Orbital of Hydrogen molecule

does anybody here know an analytical approximation of the bonding hydrogen orbital MOLECULE? I am looking for a good approximation to this orbital, that might be in some textbooks to get an ...
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812 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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425 views

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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226 views

What does the notation $|x_1,x_2\rangle$ mean?

I would like clarification on an equation in the paper "Free matter wave packet teleportation via cold-molecule dynamics", L. Fisch and G. Kurizki, Europhysics Letters 75 (2006), pp. 847-853, DOI: ...
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203 views

Why do we consider the evolution (usually in time) of a wave function?

Why do we consider evolution of a wave function and why is the evolution parameter taken as time, in QM. If we look at a simple wave function $\psi(x,t) = e^{kx - \omega t}$, $x$ is a point in ...