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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53 views

Quantum Harmonic Oscillator Virial theorem is not holding

I'm asked to calculate the average Kinetic and Potential Energies for a given state of a quantum harmonic oscillator. The state is: $$ \psi(x,0) = \left(\dfrac{4m\omega}{\pi\hbar}\right)^\frac{1}{4}e^{...
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Where do the constants in front of the analytical solution of the quantum harmonic oscillator come from?

I am going through Griffiths' Intro to QM, and in his solution the quantum harmonic oscillator, he just derived the recursion formula: $$a_{j+2}=\frac{-2(n-j)}{(j+1)(j+2)}$$ Using this, we can find ...
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Why this factor $1/r$ is used in the equation of asymptotic behavior of scattered wave?

Why $1/r$ factor is used? And in this equation $f_k(\theta,\varphi)$ is scattering amplitude then why plane wave ($e^{ikz}$) amplitude is not used?
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How and why this form of wave function is used to obtain time evolution of wave packet

In Cohen Tannoudji Book Topic Of Stationary Scattering States ( In Asymptotic Form Stationary Scattering States) For time evolution of wave packet they had used this equation: I'm not getting this ...
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Multi-mode Fock states

I would like to ask if this expression makes sense. $$|Ψ⟩ = \frac{1}{\sqrt{7}}(|1000000⟩+|0100000⟩+|0010000⟩+|0001000⟩+|0000100⟩+|0000010⟩+|0000001⟩)$$ For example, $|1000000⟩$ represents one particle ...
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Unplugging the detector in the quantum double slit experiment

I'm thinking about the quantum double slit experiment. Specifically the part where if you had a detector pointing at one of the slits, this would collapse the wave function and the interference ...
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A particle starts out in a linear combination of two stationary states [closed]

I have a doubt in finding the probability density. I would be thankful if someone helps me out in this question: $E_1$ and $E_2$ are the energies associated with $\Psi_1$ and $\Psi_2$. It follows ...
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In what cases can we show that the ground state wavefunction from the variational principle is at all similar to the true wavefunction?

The variational principle gives a ground state that minimises the expected energy from the TISE, while from this we can be sure that we have an upper bound on the ground state energy, it is not always ...
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Number of ways to arrange three spin 1/2 fermions

Consider the following two configurations $(N_{1}, N_{2}, N_{3})$ of three spin-$1/2$ fermions: $$ (2,1,0), (1,1,1) $$ where $N_{n}$ denotes the number of particles in the $n$th state. I learned the ...
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Probability density for free gaussian wave packet

I am doing an exercise where the wave function for t>0 is: $\psi (x,t) = \frac{e^{\frac{-x^2}{4a^2(1+\frac{it}{\beta })}}}{(2\pi)^{\frac{1}{4}}\sqrt{a} \sqrt{1+\frac{it}{\beta }}} $ with $\beta =...
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Bound state of Hydrogen atom at large $r$

When the radial equation of SE is solved for Hydrogen atom, to see the asymptotic behavior, we assume $r$ tends to infinity. The differential equation we are left with is: $$ d^2U/dr^2 = -\frac{2mE}{\...
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The matter wave concept, convergence in the macroscopic world and the contradiction

One of the postulates of quantum mechanics states that as the system of interest goes from the microscopic world to the macroscopic world, the quantum physical laws converge to classical physics. I ...
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Full discription of wave behaviour of particles (e.g. electron diffraction) by wavefunctions

For example, according to the result of electron diffraction experiment, particles show wave characteristics. I also heard that according to Copenhagen interpretation and decoherence theory, before ...
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Does this wave function violate any rules of Quantum mechanics?

I am studying bosons and fermions in quantum mechanics. We learned that for a two-particle system with identical bosons, the wave function can have the form (assume symmetric spin) $$ \frac{1}{\sqrt2} ...
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4-momentum operator, in geometric algebra [closed]

David Hestenes suggests that the relativistic wavefunction can be expressed in geometric algebra as follows: $$ \psi = \rho^{1/2}e^{B/2I} R \tag{2} $$ where $$ \psi \tilde{\psi}=\rho e^{B/2I} e^{-B/...
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Laughlin wave function and CFT

I have a question regarding Eq. (3.5) in Moore & Read's paper. They said \begin{equation} \Psi_{\text{Laughlin}}=\left\langle\prod_{i=1}^{N}e^{i\sqrt{q}\phi(z_i)}\exp\left[-i\int \mathrm d^2z^{\...
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Contradictory statements on product states for distinguishable particles in Quantum Mechanics

Page no. $5$ in Many-Body Theory Exposed! by Willem H Dickhoff & Dimitri Van Neck states the following: The complex vector space, relevant for N particles, can be constructed as the direct ...
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How to generalize outer product into its integral form assuming continuous basis?

(My current physics study is undergraduate quantum mechanics) By definition, the inner product is $w^Tu= \sum_i w_iu_i$, the outer product is $wu^T=w_iu_j$. According to Griffith, the inner product ...
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Finite potential whose (normalisable) wavefunction doesn't vanish at infinity

I can come up with normalisable wavefunctions which don't vanish at infinity. However, I cannot come up with a potential so that it satisfies TDSE (the examples I think of are not differentiable at ...
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624 views

Interpreting group velocity of free particle wave packet

I am trying to understand the concept of group velocity of a free particle wave packet: $$\Psi(x,t) = \frac{1}{\sqrt{2 \pi}}\int_{-\infty}^{\infty} \phi(k)e^{ikx}e^{-\frac{i \hbar k^2 t}{2m}}dk.$$ ...
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Why do physicists, in quantum mechanics, call average an expectation value, not expected value? [closed]

I guess there is a specific reason for this - calling the expected observation $$\langle\psi|\hat{Q}|\psi\rangle$$ (for a normalised wavefunction) an expectation value. I heard somewhere that, in ...
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Wheeler delayed choice experiment : why the mirror is not an interaction collapsing the wave?

I came across the Wheeler delayed choice as it is described in Wikipedia : To summerize, the individual photon has many paths and some path leads to detectors that reveals the path taken and some ...
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Expectation value and the probability of finding a particle [closed]

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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50 views

Am I understanding the scattering amplitude correctly?

Suppose we have a particle, which is described by the wave function Ψ1, which hits another particle. In the final state, we get a superposition of an incident plane wave Ψ1 and a scattered spherical ...
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44 views

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

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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Is the energy of the potential step quantized?

I'm solving the Schrödinger equation for a potential step and I was wondering if the energy of the potential step is quantized?
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Something special about energy eigenstates when it comes to time evolution?

A particle is subject to an infinite square well potential with $$V(x)= \begin{cases} 0 & −a \lt x \lt a\\ \infty & \,\,\,\,\text{otherwise} \end{cases}$$ At a time $t=0$ its ...
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Classical & weak solutions of Schrödinger equation [closed]

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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Identifying number of node of an arbitrary wave function

Given an arbitrary wave function, what is the most general way to identify number of its nodes? By arbitrary, I mean we don't have any predefined conditions (like wave function of an atom, a harmonic ...
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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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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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Meaning of the terms in Wave Equation for a Transverse Wave [duplicate]

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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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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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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Why the statement “there exist at least one bound state for negative/attractive potential” doesn't hold for 3D case?

Previously I thought this is a universal theorem, for one can prove it in the one dimensional case using variational principal. However, today I'm doing a homework considering a potential like this:$...
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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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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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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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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 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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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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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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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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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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Degeneracy of two electrons on a ring

The one-particle solution to the particle-on-a-ring problem is $\psi_m(\phi_j) = \frac{1}{\sqrt{2\pi}}\exp\left(-im \phi_j\right)$ for $m=0, \pm 1, \pm 2, \cdots$ corresponding to energies $E_m = \...
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Half-integer eigenvalues of orbital angular momentum

Why do we exclude half-integer values of the orbital angular momentum? It's clear for me that an angular momentum operator can only have integer values or half-integer values. However, it's not clear ...
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1answer
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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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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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