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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Constancy of Wronskian when potential has Finite Discontinuities
I am working on a problem where for the 1-dimensional Hamiltonian $\hat{H}=(-1/2m)(d^2/dx^2)+\mathcal{W}(x)$ with $\mathcal{W}$ assumed to be smooth and real and $\Psi$ as solution to $\hat{H}\Psi=E\...
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Infinite 3D sphere well with Dirac Delta potential function at the origin
A spinless particle of mass $m$ is constrained in a 3D region of zero potential within an impenetrable spherical shell of inner radius $r = a$, with a delta function potential at the origin given that ...
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Is there a differential equation describing the wavefunction of a hadron?
In Newtonian Physics there's a differential equation describing the motion of multiple bodies in orbit around each other. In non relativistic quantum mechanics there's a differential equation ...
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General wavefunction for a system of two coupled, quantum oscillators
Suppose we have two quantum harmonic oscillators, with different masses $m_{1},m_{2}$ and frequencies $\omega_{1,2}$. Then we can say particles are \emph{distinguishable}, in the sense that particle $...
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Probability of finding particle in ground state [closed]
A particle is in a one dimensional infinitely deep square well, from 0 $\leq x \leq L$. Find the probability of finding the particle in the ground state of the square well if the wave function for a ...
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Proof of normalization constant of wave function to be independent of time
I am trying to prove that the normalization constant is independent of time. If we have fixed it for a particular time then it will remain constant for all time.
Suppose $\psi(x,t)$ is a wavefunction.
...
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Normalize the wave function Ψ(𝑥) = 𝐴 e (–𝑘𝑥2) , A and a are constants, over the domain −∞ ≤ 𝑥 ≤ ∞ [closed]
2. Normalize the wave function Ψ(𝑥) = 𝐴 e (–𝑘𝑥2) , A and a are constants, over the
domain −∞ ≤ 𝑥 ≤ ∞
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What are critical differences between conventional lasers and matter lasers?
In what ways would a coherent stream of matter waves/De Broglie wavelength interacting with the environment differ from a conventional coherent stream of photons? What have been observed and what have ...
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How do I normalise the wavefunction of a hydrogen 1s orbital to obtain the normalisation constant?
The wavefunction I've been given for a 1s hydrogen orbital is:
$$ \Psi = A e^{-r} $$
And I need to normalize this to find the value of A. I understand to normalise this I would inset this wave ...
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Doubt in the completeness of wave function
I am reading about the completeness property of wave function. The following is given about it-
The energy eigenstates are complete in the sense that any reasonable wave function $\psi(x)$ can be ...
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Predicting the probability distribution in a potential
I've been dealing with a kind of problem in quantum mechanics, where they give us an arbitrary potential, and then ask us to predict the form of the probability amplitude or the wave function.
The ...
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What is the general solution of one-dimentional time-independent Schrodinger's equation?
As I tried to learn quantum mechanics I have found two solutions of one-dimensional time-independent schrodinger equation in various resources.
One is,$$\psi(x) = Asin(kx)+Bcos(kx)\\\text{where}, k = \...
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How does $U(x,t;x',0) \to \delta(x-x')$ as $t \to 0$ preserve the norm?
If the propagator approaches the dirac delta as the difference in time decreases then we would expect
$$\lim_{t \to 0} \langle \Psi(x,t) | \Psi(x',0) \rangle$$
$$\int_{- \infty}^{+ \infty} |\Psi(x',0|^...
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Manipulations of the wavefunction are hard to physically understand
So i am learning about quantum mechanics, still a beginner but i am already struggling to understand the manipulations of the maths involved that gives the answer.
For classical physics I can ...
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Probability density for position and momentum and the wave function
I know that if a particle has a wave function $\Psi(x)$ at a time $t$ then the probability density for the position of the particle is given by $|\Psi(x)|^2$, and if $\phi(p)$ is the Fourier transform ...
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Why is it useful to characterize relativistic particle states by this kind of wavefunction?
This question is a followup to my previous one "Why are momentum eigenstates in QFT plane waves?
" as it made sense for me to ask this separately, in a self-contained manner.
In QFT we have ...
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Assumption made for the WKB approximation in radial coordinates
I was thinking the other day, if you had the Schrodinger equation in 3-dimensions, and had a spherically symmetrical potential. Ie.:
$$-\frac{ℏ^{2}}{2m}∇^{2}ψ+V(r)ψ=Eψ$$
Then you could simplify the ...
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States with defined energy and time evolution
Consider the following simple problem:
We have a step potential:
$$V=V_0\Theta (x)$$
so the Hamiltonian is:
$$H=\frac{p^2}{2m}+V_0\Theta(x)$$
and we want to find the eigenfunctions of the Hamiltonian $...
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Phase-shifting the one-dimensional wave function
Let $\Psi(x)$ be the wave function for a one-dimensional quantum-mechanical
system, $i=\sqrt{-1}$ and let $p_x$ be the momentum operator in one dimension.
Show that for any real number ℓ one has
$$e^{...
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Born Rule proof for freshmen
Although early quantum mechanics are taught in many freshman courses, the Born Rule is almost never proved at that stage. Is it even impossible to elementarily prove that the probability density is ...
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Physical meaning of $\langle nlm|\hat{z}|n'l'm'\rangle$
I'm working on a quantum mechanics problem with some friends and we're trying to make an argument using symmetry rather than maths. What would the physical interpretation of $\langle nlm|\hat{z}|n'l'm'...
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3answers
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Solving the Schroedinger equation with the initial condition as an energy eigenstate [closed]
I was studying quantum mechanics by watching a video lecture series. In the lecture https://youtu.be/TWpyhsPAK14?list=PLUl4u3cNGP61-9PEhRognw5vryrSEVLPr&t=2784 , the professor tries to solve the ...
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Expectation value of $\frac{1}{r^2}$ of hydrogen atom [duplicate]
I'm studying on the derivation of $\left\langle \frac{1}{r^2} \right\rangle$ by using the book Nouredine Zettili. The derivation in the book is as follow:
I can understand the rest of the derivation ...
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1answer
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Why is the reflection coefficient 1 for step potentials where energy is less than the potential?
Consider a potential $V(x)$ which is zero when $x<0$ and $V_0>0$ when $x>0$. Suppose there is an incident particle with momentum $p=\hbar k$ and energy $E = \hbar^2 k^2 / 2m < V_0$ coming ...
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I am having a doubt in graphs of $4πr^2|\psi|^2$ vs $r$ and $4πr^2|R(r)|^2$ vs $r$
To show radial probability, in some sources they used the graph $4πr^2|\psi|^2$ vs $r$
In some other sources, they used the graph $4πr^2|R(r)|^2$ vs $r$
However, $\psi(r,\theta,\phi)=R(r)Y(\theta,\phi)...
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Why is the Time Independent Schrodinger Equation so important? [closed]
The main equation of Quantum Mechanics (QM) is the Schrodinger Equation (SE):
$$i\hbar\frac{\partial \psi (x,t)}{\partial t}=H(x,t)\psi(x,t)$$
Why is this equation so important? It's important because ...
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1answer
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Position of the wavefunction's collapse
When a wave function is said to "collapse" to a single point during a measurement, is there uncertainty about the point's position or is it known infinitely precisely?
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1answer
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Wave Function for a Step Potential
If we sole the TISWE, and if energy or the particle lies between 0<E<V. If we do the calculation, Transmission coefficient (T) comes out to be zero. I get that part, but why then there exist a ...
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How is it that a particle's wave function is not a real thing, yet we can still observe it?
In the double slit experiment, scientists could see an interference pattern on the back panel. However, if the wave function is purely a mathematical object, how can it be the case that some physical ...
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1answer
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Degeneracy in $N$-particle Quantum System [closed]
I was recently introduced to the concept of $N$ particle systems in Quantum Mechanics, and the concept of indistinguishable and distinguishable particles. While reading the following material online, ...
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For $\int_{x}^{x+dx} p(x) \,dx = P(x)$ which one is the correct interval, $(x,x+dx)$ or $[x,x+dx]$? [migrated]
In Physics (both in Statistical & Quantum Mechanics) when we describe the probability function of finding a particle between $x$ and $x+dx$, we write $\int_{x}^{x+dx} p(x) \,dx = P(x)$.
Here in ...
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Many worlds interpretation and probabilities
How is the many worlds interpretation (MWI) of QM consistent with the probabilistic interpretation of the wave function (given by Born's interpretation)? For example, say a particle has a 90% chance ...
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For the even wavefunction and odd wavefunction, can we estimate whether the energy of the system is positive or negative?
For the even wavefunction and odd wavefunction, can we estimate whether the energy of the system is positive or negative? And for which of (odd or even wavefunction) energy is higher? You can consider ...
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1answer
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Is there any practical potential for which first derivative of wavefunction is continuous? [duplicate]
As we know that first derivative of the wavefunction is discontinuous when the potential is infinity. Is there any practical potential for which first derivative of wavefunction is continuous?
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How would an atom look like if we knew where the electrons were? (I know it’s silly) [closed]
So we have the normal depictions of atoms, a nucleus made of spherical protons and neutrons, and orbiting spherical electrons, them we have the fuzzy blob depictions, where there is a center, and then ...
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Wave packet in quantum mechanics?
When we talk about light waves or EM waves, we simply say that the wave packet is the superposition of other waves of different wavelengths. In quantum mechanics, we say the same thing; the ...
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What is the difference between beats and wave packets?
What is the specific difference between beats and wave packets.
According to my book both are the formed by superposition of two waves having slightly different frequencies
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On using Python to solve Time Independent Schrodinger Equation, the eigenfunctions have their values “pushed” to one of the boundaries?
I am having trouble using numerical methods to solve Time Independent Schrodinger Equation. I am considering a quartic potential function: $$ V(x) = x^4 -4x^2.$$
$$
-\frac{d^2\psi(x)}{dx^2} + V(x) \...
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Doubt during derivation of Ehrenfest's theorem [duplicate]
I was reading about the derivation of Ehrenfest's theorem in this website when I came across this step:
Substituting from Schrödinger's equation (137) and simplifying, we obtain $$\frac{d\langle p\...
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3answers
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From bra & ket vectors to wave functions
I have a hard time understanding how the transition happens between the two. Starting from Schrödinger eqaution for kets:
$$i\hbar\frac{d}{dt}\left|\psi\left(t\right)\right\rangle =\hat{H}\left|\psi\...
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Tightrope or vibrating string (high school question)
The different types of waves propagating freely in the space (without damping or perturbative sources) can be described by a single equation of D'Alembert!
In the case of a taut string, the wave ...
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Probability density of a free particle
I have been recently studying QM and I have encountered the case of a free particle. I understood that a free particle travels in the form of a wave packet where we get
$$\psi (x) = \frac{\int_{-\...
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1answer
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Tensor product of wavefunctions
In a system of two non-interacting particles in a one-dimensional infinite square well, we represent the eigenstate of the whole system as the tensor product of the eigenstates of the individual ...
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Determining coefficients for wave function solutions of an electron in a periodic potential
In Kittel's Intro to solid state physics, when solving the schrodinger equation for a periodic potential, we begin by writing the potential and the wave function as fourier series of the form
$$\psi = ...
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How to obtain the wave function of time-dependent coupled two harmonic oscillators?
The form of the Hamiltonian of this system is
\begin{equation}
H = \frac{p_1^2}{2} + \frac{p_2^2}{2} + \frac{x_1^2}{2} + \frac{1}{2}\omega^2(t)x_2^2 + \frac{q}{d}x_1x_2
\end{equation}
where $p_1$ and $...
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Finite square well and continuity [duplicate]
In solving finite square well problem, we solve the TISE inside and outside the well, and we match the wave function at the boundary, by the continuity of wave function.
Now this bugs me, since the ...
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1answer
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Quantum Field to Layman
How would you describe a Quantum Field in layman's terms? Is is some function that provides information about the universe in a particular space-time region? My motivation for the latter question is ...
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Why does the minimization of the Hartree-Fock Hamiltonian consider only the complex conjugate wave function?
We have the Hartree wave function of N particles: $ \psi_{H} ( 1, ...., N) := \phi_{1}(1) \cdot \phi_{2}(2) ... \phi_{N}(N) $ where $\phi_{j}(I)$ is the one particle wave function of the I-th particle ...
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Can the initial wavefunction be discontinuous?
In a infinite potential well of width $a$, an electron starts in the left half and at $t=0$; it is equally likely to be found at any point in that region.
To find the wavefunction at later times, we ...
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2answers
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Coefficients of the wave function - a free particle in a box [closed]
If we solve the time independent Schrödinger equation for a particle in a box of length $L$, we get:
$$\psi_n\left(x\right)=A\sin\left(\frac{\pi n}{L}x\right)$$
I then see that we normalize $A$ such ...
