Questions tagged [wavefunction]
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. DO NOT USE THIS TAG for classical waves.
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How does 1D Schrödinger equation arise out of the postulated 3D Schrödinger equation and solving 1D particle using 3D Schrödinger equation?
I've stumbled upon this question when I was trying to solve the Schrödinger equation for a particle confined to a 1D line with some given time independent potential $V(x)$.
The energy eigenstates ...
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Plane waves, angular momentum, and the 2D Schrödinger equation
I've been thinking about the 2D Schrödinger equation for a free particle, particularly in polar coordinates, and the particular solution
$$\Psi(r, \phi) = J_1(r)e^{i\phi}.$$
This solution has an ...
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Normal Base for Hilbert Space of delta Potential Well
I'm interested in the problem of an attractive $\delta$ potential. The Hamiltonian is given by
$$
H = - \frac{\partial_x^2}{2m} - V \delta(x).
$$
Solving this typically entails looking at scattering ...
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If an electron is inside an atom, does the expected value of spin measurements also depend on the orbital wavefunction?
The total quantum state of an electron in an atom can be written as the product of the orbital wavefunction and a spinor representing its spin state, $\Psi = \psi(r,\theta,\phi) \otimes \chi$. Say you ...
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Definition of expectation value for momentum [duplicate]
I think this is probably a stupid question but I'm confused over how the expectation value for momentum is calculated. It is always given as $$⟨𝑝⟩ = ⟨𝜓|\hat{p}𝜓⟩ = −𝑖\hbar∫𝜓^*(𝑥)\frac{d𝜓(𝑥)}{...
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Free particle in spherical coordinates
I'm trying to solve the very simple equation:
$$-\frac{\hbar^2}{2m}\nabla^2 \psi = E\psi$$
but in polar coordinates. I used separation of variables to find out that my wave function is of the form: $$\...
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Transformation of wavefunction
While learning QM, I was wondering how would the wavefunction of a particle, suppose charged particle, look for different observers moving with respect to each other.
To begin with, let the electric ...
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The $L^2$ operator not returning the expected value (2) when applied on the (2,1,0) Hydrogen wave function [closed]
Let the Hydrogen wave function for the state $n=2, l=1, m=0$ be described as:
$$\psi_{210} =cos(\sigma )*f(r)$$
Since the squared angular momentum eigenvalue is $L^2=l(l+1)$, I would expect it to be 2 ...
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How to calculate the inner product $ \langle T|x \rangle = \langle\frac{p^2}{2\mu} |x\rangle$? [closed]
If I have a wave function $ \psi (x) = \langle x|\psi \rangle$, now I want to use kinetic energy representation $$ T = \frac{p^2}{2\mu} ,$$ where $ \mu $ is the mass of particle. I try to
\begin{align*...
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Regarding to the asymptotic solution of quantum harmonic oscillator
In quantum mechanics, the radial equation of the SHO takes the form
\begin{align}
\frac{d^2 u}{dx^2}+\left(\epsilon-x^2-\frac{l(l+1)}{x^2}\right)u=0,
\end{align}
where $x=\sqrt{\frac{m\omega}{\hbar}}r$...
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Is the spherical outgoing wave solution to the Schrodinger equation not a member of $L^2$?
I was reading a discussion about the Mott problem, where the authors discuss the outgoing spherical wave solutions to the Helmholtz equations $\nabla^2 f = - k^2 f$. This equation can also be ...
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Wavefunction with determinate momentum
In page 100 Griffiths' Introduction to Quantum Mechanics, Griffiths states that the eigenvector of $\hat{p}$ in the position basis is $\frac{1}{\sqrt {2\pi\hbar}}e^{\frac{ipx}{\hbar}}$ and states that
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Why is the time derivative of the wavefunction proportional to a linear operator on it? [closed]
I am currently trying to self-study quantum mechanics. From what I have read, it is said that knowing the wave function at some instant determines its behavior at all feature instants, I came across ...
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Proof that separation of variables leads to a complete basis of wave function in spherical coordinates [duplicate]
In griffith's introduction to quantum mechanics (chapter 4), there is an analysis of the stationary states of a particle given a potential function $V(r)$ that only depends on the radial distance $r$, ...
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When solving a paraxial Helmholtz equation ... ? Amplitude vs Wave function
Given the paraxial Helmholtz-equation (PHHE). Why are the solutions formed by the amplitude function of a wave function, but not the wave function itself? Example, a form of a wave function is defined ...
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