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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Eigenvalue and Eigenfunction for a particle trapped in a 1D infinite asymmetric potential well

As we're barely scratching the surface of Quantum Physics in class, we haven't been taught about asymmetric potential wells. However, I find it fascinating, moreover difficult, to find the eigenvalues ...
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Comparing the quantum version of the infinite well to classical version

In the quantum version of the infinite well, the energy eigenvalues can be precisely determined. The energy is all in the form of kinetic energy, $E=\frac{p^2}{2m}$, and so, classically, the magnitude ...
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Meaning of the terms in Wave Equation for a Transverse Wave

I read in the book that partial derivative of the wave function, $y(x,t)$ with respect to $x$ (here $x$ is the position of the particle on the $x$-axis) corresponds to studying the shape of the string ...
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Classical & weak solutions of Schrödinger equation

Consider the problem of an infinite square well $$ V(x) = \begin{equation} \begin{cases} 0, \qquad {\rm if}\quad0 \le x \le L \\ \infty, \qquad{\rm otherwise} \end{cases} \end{...
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What motivates the trial solution of $\left[-\frac{\hbar^2\nabla^2}{2m}+\frac{e^2y^2B^2}{2m}-\frac{i\hbar eyB}{m}\right]\psi(x,y,z)=E\psi(x,y,z).$?

The time-independent Schrodinger equation for the problem of charged particles in an uniform magnetic field ${\vec B}=B{\hat k}$, in the Coulomb gauge ${\vec A}=(-yB,0,0)$, reduces to the following ...
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Where does the expectation value of $x$ formula come from?

I want to understand precisely where the formula for the expectation value of $x$ comes from (in QM): $$\langle x\rangle=\int _{-\infty}^{\infty}\psi ^*x\psi dx $$ I know that an expectation value (in ...
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Expectation value and the probability of finding a particle

I'm trying to understand basic quantum physics, as I understand, the expectation value of some random distribution gives us the outcome that we might expect(highest probability) if the event is done ...
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Eigenfunctions and eigenvalues of particle in 2D box

A particle in a 2D potential box has two degrees of freedom. It is bound by the infinite potentials at the boundaries. Our professor asked us to resolve this into its respective $x$ and $y$ components,...
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In what way does spin-1/2 show up in regular interference experiments? [closed]

I'm trying to better understand spin-1/2 (have been for a while...) and I'm hoping for a little help cleaning up my understanding, or at least helping me to see where my line of reasoning has going ...
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How do I find the wave function in a separable Hilbert Space?

I am confused as to how I would go about finding the wave function in the Hilbert Space. As I understand, a wavefunction in the Hilbert space can be represented as $$|\Psi\rangle = \sum_{n} c_n|\psi_n\...
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The expectation value of momentum in an infinite well stationary state [duplicate]

For a particle in an infinite well potential given by: I am able to successful derive the normalized wavefunction as: $$u_n(x) =\sqrt{2/a}\sin(\frac{n\pi x}{a})$$ where this normalization has the ...
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Normalize the variational wave function

I am trying to normalize the following variational wave function: $$\psi(x,\alpha)= |x|^{\alpha} + L^{\alpha}$$ and I'm using this: $$1= \int_{-L}^{L} |\psi(x,\alpha)|^2 dx$$ Solving the integral gave ...
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Why wave function does not vanish at a Dirac delta potential?

I have studied that a wave function should vanish at the location of an infinite potential. Consider a direct Delta delta potential at $x=0$. Why does does function not become zero here at $x=0$?
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What is the physical meaning of multiplication of two wavefunctions?

In the amount of quantum mechanics I'vs learnt I understand what wave functions are, how do we extract information from them and so on, and that addition of two wavefunctions on renormalization gives ...
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Degeneracy of energy eigenstates when $E > V$

Reading my text book on quantum physics, I found the following statement: Let's suppose we have a short-scale force, so we have a potential energy such that $$ V(x) = V_- \quad \text{if} \quad x \ll 0 ...
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Weird quantum linear operator [closed]

For a problem sheet at uni, I need to find eigenvalues and normalised eigenstates of a linear operator. This operator is $\hat{Q}$ and is defined by its action on the normalised eigenstates of the ...
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What do atomic orbitals represent in quantum mechanics?

I am learning the basics of quantum mechanics and am familiar with the Schrödinger equation and its solution, but I was confused about what the familiar atomic orbital shapes represent? Do they ...
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What does a linear combination of eigenfunctions represent and how do I calculate energy for such a system? [closed]

I tried to solve the question in the image. I did go through some literature and have included that text too. My problems are, how do I calculate the energy? , does this represent a superposition of ...
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Why is the following property regarding the general solution of the Schr. Eq. valid only for the free-particle case?

In my quantum mechanics lecture notes(see picture at the bottom), they say the plane wave basis $\{\phi_k(x)=\frac{1}{(2\pi)^{3/2}}\exp(ik\cdot x)\}$ is so general that any $ \psi \in L^2(\mathbb{R}^3)...
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28 views

Plane wave basis in a box. Inconsistent terminology about Fourier series and transform

I will refer to the one-dimensional equivalent of the problem in the picture for simplicity. At the end of the picture, the author argues that the Fourier transform of the function $\psi(x)$ is the ...
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Double slit: does some electron stop at the slit material?

I have a very simple question about the double slit experiment with electrons. Does some electron does not pass through the slits at all? All explanations I read about the experiment demonstrate the ...
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Parallels between Tight-binding wavefunctions and Bloch states

I was reading Robert Knox's Theory of Excitons and I came across a certain statement that I have trouble reconciling, both mathematically and conceptually. The context concerns the electronic ...
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Does the Schrodinger equation obey the rule for differentiating a function if the function is in terms of the wavefunction?

Does the Hamiltonian operator act like a derivative when acting on a functional in terms of wavefunctions? For example, does $$H\psi^2=2\psi H\psi$$ hold true? More generally, if the functional, $F(\...
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Question regarding step potential

We are learning about step potential in class. I have completely understood that the behavior of the wave function representing the particle, can have different responses depending on the energy of ...
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58 views

Hydrogen Atom in two spatial dimensions with $1/r$ potential [closed]

I am almost new to Quantum Mechanics. Recently I learned about the hydrogen atom in three dimensions. I struggle to answer the following exercise where the hydrogen atom in two dimensions is ...
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Why Semiclassical Approximation is called “semiclassical”? [closed]

I am studying the Semiclassical Approximation (WKB Approximation) in the context of Quantum Mechanics. It seems to me just another way to approximate the solution of a differential equation, with ...
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Mathematics behind particle on a ring in QM [duplicate]

In QM particle on a ring, the Schrodinger in cylindrical coordinates is given by $$\frac{d^2\psi}{d\phi^2}=\frac{-2IE}{\hbar^2}\psi$$ where $I$ is the moment of inertia. Setting $ k^2=2IE/\hbar^2$, ...
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For energy eigenstates $\psi(x)$, does $\psi(x)\to0$ as $|x|\to\infty$ imply $E<V(\pm\infty)$ and vice-versa? If so, how can it be proved?

Let $\psi(x)$ be a solution of the time-independent Schrodinger equation (TISE) in one-dimension $$\psi''(x)+\gamma\left[E-V(x)\right]\psi(x)=0$$ where $\psi(x)$ is called an energy eigenfunction, $\...
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Is there any “wave equation” analogous to Schrodinger’s Equation in QFT?

As in, is there any sort of partial differential equation, for let’s say, a real scalar field qft theory that you could solve for a wave function that describes the qft?
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Boundary condition (periodic) of many body wave function?

As we know, in a lattice system, the boundary condition for one body wave function is $\psi(N+1)=\psi(1)$. However, what is the exact periodic boundary condition for many-body wave function? To ...
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Can average momentum be imaginary?

I am new to quantum physics. We just learnt about wave equations, observables and expectation values today. What really caught my attention was the expectation value of average momentum and energy: $$\...
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How does the barrier affects quantum tunneling?

I am picturing a sine wave acting as a barrier for my quantum tunneling experiment, it can be an electric field etc. Now I have an electron whereby a portion of its probability intersects and even ...
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Is the universal wave function expressible? How does it relate to energy in GR? What is it compatible with?

I am trying to wrap my mind around the universal wave function. Which doesn't seem like a clear concept to me. Is there an actual equation which is the universal wave function or is it inferred from ...
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52 views

Wave function for particle in an infinite well located at $a$ and $b$ [closed]

I know that the wave function for a particle in a infinite potential well located between $0$ and $L$ is: $$\psi_n = \sqrt{\frac{2}{L}}\sin\frac{n\pi x}{L}$$ But I don't have idea how to apply ...
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Why are energy eigenstates required to be bounded?

The MIT lecture notes for Quantum Physics-II course says that for a solution $\psi(x)$ (to the time-independent Schrödinger Equation) to be acceptable, it is required to be continuous and bounded, and ...
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Wave equation solution satisfied? [closed]

I'm having a problem trying to show that this solution satisfies the wave equation. I discovered this solution given by $$\psi(x,t)=e^{-(ax-bt)^2}$$ but I'm stuck trying to prove that solution ...
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68 views

Can one derive the Schrödinger equation from probability density arguments?

My interest is regarding a probability amplitude $\psi^\dagger \psi: \mathbb{R}^2 \to [0,\infty)$, where $\psi:\mathbb{R}^2 \to \mathbb{C}$. The average position is $$ \bar{x}=\int_{-\infty}^\infty x (...
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What does “singlet state” mean in the context of colour charge, and do red, green and blue colours cancel?

I find myself very confused by the usage of spin terminology to other quantum numbers. A singlet state is the only spinless state of the system. Now, if we consider the possible colour states $|r\...
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How to find probability of finding a particle outside the region in which it is confined?

Problem: Consider a one-dimensional particle of mass $m$ which is confined within the region $0 \le x \le a$ by a potential $V(x)$ and whose wave function is $\Psi(x, t) = \sin(\pi x/a)\exp(−i\omega t)...
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Hermiticity of the Hamiltonian operator with probability conservation

I am following MIT lessons on quantum physics (Prof. Zwiebach): Part I, Lecture 6, at https://ocw.mit.edu/courses/physics/8-04-quantum-physics-i-spring-2016/lecture-notes/ Video lecture: https://youtu....
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How to get eigenvalue equation with Transfer Matrix Method?

I'm studying a paper on physics entitled "Bound State of the One-Dimensional Dirac Equation for Scalar and Vector Double Square Well Potential" It states in the paper that "It is ...
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Position matrix representation in QM [closed]

For quantum harmonic oscillator, if wave function is in a superposition of two wave functions, $$ \psi(x)=(1/\sqrt2)\psi_n(x)+(1/\sqrt2)\psi_m(x) $$ and position operator is represented as a matrix, ...
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When is a logarithm of the wavefunction well-defined?

It is sometimes convenient to write the wavefunction as $$ \Psi(x,t)~=~ e^{\Phi(x,t)} $$ and then work with $\Phi$ instead. This is particularly sensible in the context of the WKB approximation, where ...
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Meaning of wave function squared, notational confusion

For a non-degenerate ground state in a system with $N$ electrons, we may write the wave function as, \begin{equation} \psi(r_1, r_2,..., r_N) \end{equation} Where the $r_i$ represent the position of ...
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Show that for the QM harmonic oscillator $\langle m |x^3 |n\rangle =0$ [closed]

I have to demonstrate the following result: $$ \langle m |x^3 |n\rangle = \int_{-\infty}^{+\infty}\Psi_m x^3 \Psi_n =0 $$ Unless $m=n-3$, $m=n-1$, $m=n+1$, or $m=n+3$. I tried using the normalized ...
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What happens if we expand out fields in terms of different functions?

When we "expand" our classical fields, for example the Dirac field, in the standard way which we later go on to "quantise": $$\psi(x,t)=\int d^3\tilde k \sum_{a=1,2}\left(b_a(k)u^a(...
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37 views

Regions separated by infinite potential

In normal (i.e. finite potential) QM systems, a particle wave function can have many nodes with $| \psi \rangle \neq | 0 \rangle \land \langle x | \psi \rangle = \psi(x) = 0$. However, in systems that ...
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173 views

Time-dependent Schrödinger equation of the harmonic oscillator

hello I have the time dependent Schrödinger equation of a harmonic oscillator $$i \hbar \frac{\partial}{\partial t} \Psi(q,t) = - \frac{\hbar^2}{2M} \frac{\partial^2}{\partial q^2} \Psi(q,t) + \frac{M ...
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How is the probability of finding an electron zero on its motion in a confined box?

Let's say we've put an electron in a 1-dimensional region such that the potential is $$V(x)=\cases{+k&for $x\le0$ and $x\ge L$\\0&for $0<x<L$}$$ with $k>0$. Now since electrons ...
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81 views

What is the meaning of probabilities in quantum mechanics?

In quantum mechanics, probabilities are associated with the detection of a physical event by a macroscopic device, or are events at the microscopic level also probabilistic? For example, the ...

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