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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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Relativistic hydrogen wave function for python/mathematica

Is there any library/module for mathematica or python that has the exact solution of the Dirac equation for a hydrogen atom implemented? Sympy has the wave functions the non-relativistic wave ...
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1answer
30 views

False solution of Landau Hamiltonian

The Landau Hamiltonian in 2D is given (in natural units $q=c=2m=1$) by $$ \hat{H} = (\hat{\vec{p}}-\vec{A}(\hat{\vec{x}}))^2 \,,$$ where $\vec{A}$ is the magnetic vector potential field. We know that ...
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2answers
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About the symmetric spatial part of a two-electron wavefunction: Can it be that $r_1= r_2$ less favoured than $|r_1-r_2|\neq 0$?

The two-electron wavefunction of the ground state of helium is $$ \psi(r_1,r_2)=\phi_{1s}(r_1)\phi_{1s}(r_2)\otimes (|\uparrow_1\downarrow_2-\downarrow_2\uparrow_1\rangle)/\sqrt{2} $$ where $\phi_{1s}...
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1answer
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Momentum of One Particle in a Two Particle System [on hold]

The Question: Imagine two particles in a box such that $-1<x_i<1$ where $x_i$ is the position of particle i. If the wave equation that describes this system is $\Psi(x_1,x_2) = Ae^{ik(x_1-x_2)}...
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4answers
71 views

If a wavefunction is normalized at $(x,t)$, is it also normalized at $(x\!-\!ct,t)$?

If $\Bigl( \Psi(x,t),\Psi(x,t) \Bigr) =1$, I want to find out if $\Bigl( \Psi(x-ct,t),\Psi(x-ct,t) \Bigr) =1$. My attempt was $\Bigl( \Psi(x-ct,t),\Psi(x-ct,t) \Bigr) = \int_{-\infty}^\infty |\Psi(x-...
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5answers
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Confusion about ket states and bra with position

I am very confused about the bra-ket notation of states and the fact that $$\psi(x) = ⟨x|\psi⟩$$ and $$⟨x|x'⟩ = \delta(x-x')$$ are true. What does this mean? What is the ket $|x⟩$, is it just some ...
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3answers
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What are the experiments performed to determine the position of an electron inside an atom to verify the probability wave function data?

What are the experiments performed to determine the position of an electron inside the atom to verify the probability wave function data? Is it possible to do those experiments in real life?
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3answers
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What are wave-functions in QM. corresponding to?

I was learning about a particle trapped in a double well potential $$V(0) = \infty, V(x1)=\infty$$ which can be described by $\psi_n$ for n=0,1,...,$\infty$ with corresponding Energies $E_n$. Just ...
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0answers
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Quantum Harmonic oscillator exercise [closed]

Our teacher gave us a question for exercise. I tried but I couldn't solve it. Please help me solve this problem. This section come after infinite square well on the griffits introduction quantum ...
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2answers
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Conceptual understanding of Schrödinger equation

So I followed this lecture: https://www.youtube.com/watch?v=qu-jyrwW6hw which starts of with the statement: If you have a Schrödinger equation for an energy eigenstate you have $$-\frac{\hbar}{2m}...
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1answer
104 views

Infinite square well: wall with infinitesimal thickness

Given an infinite square well, it doesn't matter how thick the wall is, the particle is trapped inside the two walls. If we make the wall of arbitrarily small but finite thickness, the particle is ...
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1answer
46 views

Scaling Problem with Variational Method

$\def\braket#1{\langle#1\rangle}$ I am attempting to solve a particular Hamiltonian by variational method. The wavefunction that I have selected is as follows: $$ \Psi = Ne^{\frac{-kr}{2}}\sum_{i=0}...
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0answers
34 views

Why does the integration over momentum has normalization constant of volume?

If I Fourier transform a wave function in position space, integration carries no normalization constant: $$\displaystyle{\phi(k) \equiv \langle k|\psi\rangle = \sum\limits_x \langle k|x\rangle\langle ...
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1answer
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How to propagate Heller's model of the Gaussian Wave Packet? [closed]

I'm working on an undergraduate research project on gaussian wave packets, but am a pretty big noob in terms of theoretical chemistry and the like. I'm having a lot of difficulty grasping this concept ...
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1answer
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Does the wavefunction probabilities have to sum to 1? [duplicate]

In quantum mechanics we are often told that $\int |\psi(x,t)|^2 dx^3 =1$. i.e. the probabilities have to sum to 1. And that this implies the time evolution operator is unitary. But can't we define ...
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1answer
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Numerical approximation of the wavefunction in a delta-potential [closed]

I am trying to approximate the wavefunction of a particle in a delta potential $U(x) = -U_0 \delta(x)$ with $V_0 \gt 0$. I am using the following formula to calculate the wavefunction: $\psi(x+\Delta ...
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0answers
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Using Expectation values to calculate Uncertanity Principle for radial distance

By starting from expectation values, find the uncertainty in radial distance for an electron in state n = 2, l = 1, m = 1. Calculate the percentage of this uncertainty with respect to the actual ...
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4answers
148 views

Does the Schrodinger wave function associated with a non-moving free particle change in time?

I'm a bit confused by an answer given on this question. In the answer with the animation of a moving free (chargeless) particle and a non-moving free particle (or a free particle with a non-zero ...
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1answer
58 views

Expectation value for the superposition of the two states (meaning of the imaginary part)

The wave function $\Psi$ of an electron that can exist in both states $n$ and $m$ is $$ \Psi = a\Psi_n + b\Psi_m \tag{6.28} $$ where $a^*a$ is the probability that the electron is in state $n$ and ...
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2answers
72 views

How is the wave function Lebesgue integrable?

Let's assume we have a plane wave $\psi(x,t)= A_{0}e^{i(kx-wt)}$ in position space. To find the momentum representation of this wave we'd apply the Fourier transform. However, I don't see how this is ...
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4answers
492 views

What is the difference between a Hilbert space of state vectors and a Hilbert space of square integrable wave functions?

I'm taking a course on quantum mechanics and I'm getting to the part where some of the mathematical foundations are being formulated more rigorously. However when it comes to Hilbert spaces, I'm ...
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1answer
55 views

Where did mess up while calculating the expected value of the momentum squared?

I have the correct answer except with a negative sign. The wave function is given as, $$\Phi=A\exp\left[-a\left(\frac{mx^2}{\hbar} + it\right)\right]$$ By squaring the momentum quantity, I found ...
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1answer
263 views

Wave particle duality and gravity

Is a particle's center of gravity at the center of its wave function or is it where we would measure the particle to be? When we measure a particle does its center of gravity shift to where the ...
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1answer
60 views

Help me make sense of the spectrum for the quantum wave function of an infinitely hard equilateral triangle

I'm trying to solve the spectrum for a equilateral Tetrahedron with infinitely hard walls. My first guess is to sum up a infinite amount of separable solutions to match the boundary conditions on the ...
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2answers
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Non-Relativistic Limit of Klein-Gordon Probability Density

In the lecture notes accompanying an introductory course in relativistic quantum mechanics, the Klein-Gordon probability density and current are defined as: $$ \begin{eqnarray} P & = & \dfrac{...
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3answers
72 views

Does the wave function carry energy and momenta?

I was reading a forum regarding the reality of the wave function, and one user (who apparently works in quantum foundations) stated that "the wavefunction carries no energy or momenta and this ...
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4answers
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Action of momentum operator on wavefunction in momentum space

In a previous question How to get the position operator in the momentum representation from knowing the momentum operator in the position representation? it was mentioned that $$\begin{align} \...
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2answers
59 views

Linear combination of 2 spherical harmonic functions

The task is to form 2 linear combinations out of the 2 given spherical harmonic functions. I dont understand why the resultant wave function has to be multiplied with the constant $1/sqrt(2)$?
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3answers
59 views

Force from a wavefunction/superposition on other particles

How does force-interaction which is dependent on some uncertain property(like position or velocity) work in QM? If I have a charged particle described by some position wavefunction, how would I expect ...
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Griffiths quantum mechanics evidence for a square-integrable solution must decay faster than $1/\sqrt{x}$ [duplicate]

In chapter 1.4 Normalization of Griffiths QM, the footnote of quote "Physically realizable states correspond to the square-integrable solutions to Schrodinger's equation" states that, Evidently $$ \...
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1answer
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Finding total flux of probability current through a sphere

For a wavefunction: $$\Psi(\textbf{x}) = e^{ikz} + \dfrac{f(\theta)}{r}e^{ikr}$$ Where $z = r\cos(\theta)$. The probability current $J$ is then given by: $$J(\textbf{x}) = J_1(\textbf{x}) + J_2(\...
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1answer
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The Pauli exclusion principle and the Pfaffian

We are talking about spinless fermion many-body wavefunctions. The determinant is a very nice structure for the Pauli exclusion principle, this is because when two single-particle states are the same,...
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6answers
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What is the probability for an electron of an atom on Earth to lie outside the galaxy?

In this youtube video it is claimed that electrons orbit their atom's nucleus not in well-known fixed orbits, but within "clouds of probability", i.e., spaces around the nucleus where they can lie ...
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Prediction and confirmation of prediction of peak of $|\Psi(x)|$

Let $$\Psi(x) = \int_{-\infty}^{+\infty}dk\cdot\Phi(k)\cdot e^{ikx}$$ and $$\Phi(k) = e^{-L^2(k-k_0)^2}$$ By the stationary phase argument, $|\Psi(x)|$ should be maximum when the phase $e^{ikx}$ ...
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2answers
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Probability to get an Eigenvalue of Angular Momentum Operator on an Arbitrary Ket

Hello physics SE community, I am currently working on Principles of Quantum Mechanics by Shankar and i get stuck in page 336 (its not even an exercise). It basically said that "we may expand any $\...
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4answers
87 views

Does a particle with infinite energy escape an infinite well?

Currently, my modern physics class is going over particles in finite and infinite wells, general quantum formalism, and tunneling. What happens to a particle as it gains an infinite amount of energy? ...
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0answers
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Can we ever “measure” a quantum field at a given point?

In quantum field theory, all particles are "excitations" of their corresponding fields. Is it possible to somehow "measure" the "value" of such quantum fields at any point in the space (like what is ...
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Approximate probability of a person's wavefunction collapsing to the moon?

As a sort of introductory concept to quantum mechanics, I've heard many explain that there's a small but nonzero probability of unlikely events happening: your hand quantum tunnels through the desk, ...
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3answers
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Why we often approximate a wave function of a particle to Gaussian wave function?

I was solving problem of two particle system. We were taking wave function generally $\psi$. Later we approximated this wavefunction of two-particle system to double Gaussian wave function. My ...
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1answer
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Relfection and transmission coefficients for wave function in $\delta$-potential

Let's assume we have some one-dimensional Delta-potential $V(x)=V_0 \delta(x)$. Then I have found numerous problems where the approach for a wave function is $$\varphi(x)=\begin{cases}e^{ikx}+re^{-ikx}...
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2answers
39 views

Indistinguishable particles and symmetrization of wavefunction

For 2 indistinguishable particles, we take the wave function to be $$\psi\pm (r_1,r_2) = A[\psi_a (r1)\psi_b (r2) \pm \psi_b (r1)\psi_a (r2) ]$$ where fermions get a - sign and bosons get a + But, if ...
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1answer
37 views

Gram-Schmidt process and degenerate subspace of the solutions to the Schrodinger's equation

So I know that in QM each linear combination of a degenerate set of wavefunctions is also a solution to the Schrodinger's equation (SE). The degenerate wavefunctions must be orthogonal to the non-...
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1answer
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Reason for $(2 \pi \hbar)^{-\frac{3}{2}}$ prefactor for quantum mechanical wavepacket

My textbook states that the prefactor $(2 \pi \hbar)^{-\frac{3}{2}}$ is not required for the following superpositioned wave function, but should be included for practical reasons without stating what ...
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1answer
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Time evolution of a free particle with a given initial state [closed]

My homework problem reads: Consider a free particle in one dimension. Write an expression for the wavefunction $\psi(x, t)$ given an initial state $\psi_0(x) = Ae^{-ax^2}$ at $t = 0$, where $A$ is ...
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2answers
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Why do we get wrong answers for orbital angular momentum if we solve it algebraically?

It is a well-known fact that the values for the square of the orbital angular momentum of a particle $L^2$ and it's projection in the $z$-direction $L_z$ are $m\hbar$ and $l(l+1)\hbar$ and that $l$ ...
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0answers
57 views

Proof that 1D bound states are non-degenerate without Wave Mechanics

Most proofs I see that all bound states of a quantum system are non-degenerate use the wave-function representation. How can we prove it using only Dirac notation, without resorting to the wave ...
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2answers
262 views

Matrix elements of the free particle Hamiltonian

The Hamiltonian of a free particle is $\hat H = \frac{\hat p^2}{2m}$, in position representation $$ \hat H = -\frac{\hbar^2}{2m} \Delta \;. $$ Now consider two wave functions $\psi_1(x)$ and $\psi_2(x)...
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2answers
54 views

Help understanding the solution of the Schrodinger equation in a finite well [closed]

Given a finite well like this I found that the TISE has the general solutions. $$\psi_{1}(x)=A_{+}e^{ikx}+A_{-}e^{-ikx} \qquad x\in[-L,L] $$ $$\psi_{2}(x)=B_{+}e^{k^{'}x}+B_{-}e^{-k^{'}x} \qquad x\in(-...
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Would the time dependent wave equation of a free particle be the same in 4D Euclidean spacetime as it is in 4D Minkowski spacetime?

In 4d minkowski spacetime the time dependent wave equation of a free particle is $$\Psi(x,t) = \left({a \over a + i\hbar t/m}\right)^{3/2} e^{- {x^2\over 2(a + i\hbar t/m)} }.$$ I notice that there ...