# Tagged Questions

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### Quick question on sketching wavefunction in well

Usually for an infinite well, the sketch for n=3 level is this: Now I think if one side of the potential barrier is higher, the particle will be more likely to spend time on the left side than ...
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### Stationary state of time-independent Schroedinger equation is always real valued function?

I am reflecting on the solution of the time-independent Schroedinger equation. My reasoning is that the stationary state of the time-independent Schroedinger equation must be a real valued function ...
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### On use of Hamiltonians for Helium

The Hamiltonian of helium can be expressed as the sum of two hydrogen Hamiltonians and that of the Coulomb interaction of two electrons. $$\hat H = \hat H_1 + \hat H_2 + \hat H_{1,2}.$$ The wave ...
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### Is it possible to find the hydrogen atom's radial wavefunctions?

Is there a way to actually find the equation of $R(r)$ without looking at a table with these equations already given? I'm given $n$, $\ell$, and $m$.
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### Kronig-Penney model

I am studying the Kronig-Penney model as treated in the book by Kittel: Introduction to Solid State Physics. In this model one considers a period potential which is zero in the region $[0,a]$ ...
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### Obtain the eigenfunction of Jz for the wave function of an electron in a hydrogen atom? [closed]

The wave function of an electron in a hydrogen atom is given by Is this wave function an eigenfunction of Jz , the z-component of the electron’s total angular momentum? If yes, find the ...
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### Expectation value of Hamiltonian in different pictures of quantum mechanics

We start with the familiar Schrodinger equation: $$i\hbar \frac{\partial \left|\psi_S\right\rangle}{\partial t} = \hat{H}_S \left|\psi_S\right\rangle$$ As we switch to a different picture than ...
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### When can we assume that the wavefunction is separable

While working out the stationary states of a single particle in a 3d infinite potential box ($V=0$ inside a cuboid of known dimensions, $V=\infty$ everywhere else), I realized I had to assume the ...
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### How will a particle with energy less than $V_{\rm min}$ behave?

Consider e.g. the finite square well: $V = -V_o$ between $x=-a$ and $x=a$, $V=0$ elsewhere Now for scattering states, $E$ must be $> 0$. For normalizable bound states, $E$ must be $< 0$ and ...
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### Solving the 1D Schrodinger Equation for a Free Particle - Two Different Methods?

So starting from the time dependent schrodinger equation I perform separation of variables and obtain a time and spatial part. The spatial part is in effect the time independent schrodinger equation. ...
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### Free particle Schrödinger Equation

Some sources give the free-particle solution to Schrödinger equation as $$\psi(x,t) =Ae^{i(kx-\omega t)} + Be^{-i(kx+\omega t)}$$ while some sources give it as $$\psi(x,t) =Ae^{i(kx-\omega t)}$$ ...
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### Regarding derivation of Probability Current

The question for the full derivation of Probability Conservation -> Probability Current was already asked here: Probability current. I apologize for not retyping it out, but it's already beautifully ...
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### Schrödinger's Equation and the depth of a finite potential well

Before I ask my question, I have to stress: I have absolutely no idea what the math is going on. I've read my textbook, several Wikipedia articles, scoured the internet, and don't feel anymore ...
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### Box normalisation and Particle in a box - Quantum Mechanics

I have been long itched by this issue of subtle difference between box-normalised free particle and infinite-dimensional potential well. Choosing a one dimensional case, the Hamiltonian in two cases ...
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### Can we safely assume $\Psi(x,t) = \psi(x)e^{-i\omega t}$ always in QM?

In the particle in a box, harmonic oscillator and in Hydrogen Atom, we can safely assume $$\Psi(x,t) = \psi(x)e^{-i\omega t}.$$ So why not make it a postulate to consider the wave function to be ...
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### Example of the time-independent Schrödinger equation having a complex solution?

We know $\Psi(x,t)$ is complex, but can $\Psi(x)$ be complex? I have seen particle in a box, well and harmonic oscillator. All have real solutions for time-independent Schrödinger equation. Hence, I ...
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### Comparison of 1D and 3D wave functions

When discussing the Schroedinger equation in spherical coordinates, it is standard practice in QM handbooks to point out that the radial part of the 3-dimensional wave equation bears a strong analogy ...
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### General wavefunction and Schrödinger Equation

I'm starting with quantum mechanics and the book I follow (Griffiths) first introduces the wavefunction as the probability density of the position of a 0-spin single particle. Later on I've realized ...
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### Wave packets and the derivation of Schrodinger's equation

I studied in my class, that a plane progressive wave cannot be used to represent the wave nature of a particle as it is not square integrable. Also, the phase velocity can get above the value of $c$, ...
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### A few parity questions for simple harmonic oscillator

I think I understand that the solution to the Schrodinger equation for the SHO is based on the Hermite polynomials (and the Guassian function). The solution set of all even Hermite polynomials are a ...
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### Formulation and probability of a wave-function [closed]

I have got this problem where I have been given the following wave function: $$\Psi = 0\quad\text{if}~|x| > a\quad\text{and}\quad A(a^2-x^2)\quad \text{if} \quad |x|< a$$ Now the first question ...
A wave function is an infinite dimensional vector space, how can it "live" in $\mathbb{R}^3$? Given the equation that is built like: $$\Psi (x,t) = \sum ^{\infty} _{n=1} c_n \psi _n (x) e^{-i E_n t / ... 5answers 715 views ### Infinite Wells and Delta Functions In considering a delta potential barrier in an infinite well, I can just enforce continuity at the potential barrier-it doesn't have to go to zero. Why then does it need to go to zero at the walls of ... 1answer 317 views ### Why is the wave function complex? [duplicate] Why should an equation (TDSE) in which first time derivative is related to second space derivative have a solution that contains i?The wave function is supposed to be complex, but I am unable to ... 1answer 192 views ### In Schrödinger equation, can we get \Psi if we know \varphi(n)? Let A be a mechanical quantity and we know its eigenfunction \varphi(n). Can we get its wave function \Psi, if we measure A for many (maybe infinite) times at the same time? 1answer 291 views ### Tunnelling through a Dirac potential barrier I am reading a QM book by Griffiths, which says it is possible for wave particle to tunnel through a barrier formulated by a Dirac function. This function is known to peak at infinity and also ... 2answers 147 views ### Does Schrödinger equation have dual-property with Heat equation? I have experimental data that Schödinger equation maintains high frequencies, while heat equation low. Does Schrödinger equation have some duality property with heat equation? 1answer 1k views ### How to find the wavefunction that solves an infinite square well with a delta function well in the middle? Solutions for the wavefunction in an infinite square well with a delta function barrier in the middle are easily found online (see here for an example). I am wondering what the wavefunction is for an ... 1answer 214 views ### Quantum harmonic Oscillator analytic method I'm using a book from Griffiths, I got really stuck about how he arrived at the approximate solution, is it just by trying( trial solution method?), I really appreciate any help on this. ... 1answer 241 views ### Given wave function at t=0, what is the process of deriving time dependent wave equation? [closed] Suppose$$\Psi (x, t=0)=Ae^{i\alpha _1}\psi _1(x)+Be^{i\alpha _2}\psi_2(x)+Ce^{i\alpha _3}\psi_3(x).If \psi _n are the energy eigenfunctions how would I derive \Psi (x,t)? I am having trouble ... 3answers 140 views ### Non-normalizable QM bound state in 4 spatial dimensions? Edit 26/Sept/13: Fixed Typo in potential I'm solving the following (seemingly simple) quantum-mechanical problem in four spatial dimensions. In natural units (\hbar^2/2m=1), the Schrödinger ... 3answers 527 views ### When Eigenfunctions/Wavefunctions are real? When the Hamiltonian is Hermitian(i,e. beyond the effective mass approximation), generally under which conditions the eigenfunctions/wavefunctions are real? What happens in 1D case like the finite ... 1answer 700 views ### How to prove dp/dt = -dV/dx? Quantum mechanics [closed] I got this problem from a book called Introduction to quantum mechanics, griffin 2nd edition. and I did not get why the solution says first term integrates to zero, integration by parts twice?! ... 1answer 320 views ### A Simple Explanation for the Schrödinger Equation and Model of Atom? [closed] I tried reading the Wikipedia article to no avail - I simply cannot understand the Schrödinger Equation (what does each of the variables mean, especially the wave function), and the Schrödinger Model ... 2answers 645 views ### Wavefunction as a combination of two stationary states - how to find those states? Lets say we have a particle in a infinite square well which has a wavefunction like this (A is some constant and d is the width of the well): \begin{align} A\left[ \sin \left(\frac{2 \pi ... 2answers 1k views ### The Energy Eigenvalue of a Wavefunction I've been reading an introduction to quantum mechanics online, and while constructing the Schrodinger equation for a free particle, the equation i\hbar \frac{d \Psi}{dt}=\hbar\omega\Psi is obtained. ... 1answer 1k views ### Variational Derivation of Schrodinger Equation In reading Weinstock's Calculus of Variations, on pages 261 - 262 he explains how Schrodinger apparently first derived the Schrodinger equation from variational principles. Unfortunately I don't ... 1answer 503 views ### Solution of 1-D Schrodinger equation for the potential V(x) = -\frac{1}{|x|} May be this question might have already been asked but i couldn't find it, so let me know if its already there. Consider a potential, V(x) = -\frac{1}{|x|} and if we apply this to a one dimensional ... 1answer 634 views ### Solving a time independent Schrödinger equation with a given potential I'm trying to rework this old homework problem, and I am having problems arriving at the same solution on the answer sheet: LetV(x)=\begin{cases}\infty &\text{ if } x < 0\\ ...
Can you help me with this problem from Sakurai: A particle of mass m in one dimension is bound to a fixed center by an attractive delta-function potential: V(x) ~= ~-a\delta(x) , \qquad ...