Partial differential equation which describes the time evolution of the wavefunction of a quantum system. It is one of the first and most fundamental equations of quantum mechanics.

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First-order and second-order wave equations, versus the uncertainty principle

In classical physics, we have second-order equations like Newton's laws, so we need to specify both position (zeroth order) and velocity (first order) of a particle as initial conditions, in order to ...
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34 views

Writing Schrodinger equation with central potential in Atomic unit

I'm struggling to write Schrodinger equation with a central potential in Atomic unit. $$ ...
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62 views

Justification of discrete spectrum for V(x) unbounded at $\pm \infty$ in Pauling and Wilson

In Pauling and Wilson, Introduction to Quantum Mechanics, they offer the following intuitive reason for the discrete spectrum of a potential which is unbounded at $\pm \infty$: This is ...
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Solution of the time independent Schrödinger wave equation using Green function [closed]

Green function are famous for solving inhomogeneous differential equations. However in some examples it is also used for solving the homogeneous differential equations. So is there any way that we can ...
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1answer
58 views

Proving that the electronic Schrödinger equation has no closed analytic solutions for >1 electron

It is stated in many books that analytic closed solutions to the time-independent electronic Schrödinger equation, $$\hat{H}\Psi = E\Psi, $$ exist for the one-electron problem (e.g. hydrogen atom, ...
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1answer
78 views

Is there some quantum potential producing exponential eigenvalues?

Usual central potentials produce quantum spectra with energy levels going as $n$, $n^2$, $n^3$ and so on, being $n$ the quantum number of the orbit. In the other extreme we have "dirac-delta" ...
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Does Heisenberg equation of motion imply the Schrodinger equation for evolution operator?

Let us choose to postulate (e.g. considering the analogy of the Hamiltonian being a generator of time evolution in classical mechanics) $$ i\hbar \frac{d\hat{U}}{dt}=\hat{H}\hat{U}\tag{1} $$ where ...
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24 views

Computational package to find the ground state of a particle in 3D domain

I am developing a numerical algorithm to find the ground state of a Hermitian matrix. Obvious applications are quantum many-body systems and particles in various potentials. I am a little stuck with ...
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3answers
75 views

Solving the Schrödinger equation where the initial wave function is an energy eigenfunction

I was watching Allan Adams' lecture on energy eigenfunctions, and there's one part (around 43 minutes into the lecture) that confuses me. Suppose we have the initial wave function $\Psi (x,0)$ such ...
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2answers
49 views

What is the energy operator and from where do we get it?

I am trying to learn Quantum mechanics from MIT OCW Videos about quantum mechanics. I have reached the 5th lecture. Please help me in understanding this: In the middle (At 32:08), the professor wrote ...
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1answer
70 views

Poles for a particle scattered in a delta potential

I am working on problem a professor gave me to get an idea for the research he does, and have hit a point where I'm having a difficult time seeing where I need to go from where I'm at. I would also ...
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1answer
91 views

Divergent solution in time-dependent Schrödinger equation

if I transform the time-dependent Schrödinger equation without a potential I get: $$ - \hbar \omega \psi(\omega,x) = \frac{- \hbar^2}{2m} \frac{\partial^2 \psi(\omega,x)}{\partial x^2}$$ The ...
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79 views

Initial condition for Fourier transformed Schrödinger equation

I asked in this thread Time-dependet Schrödinger equation how to solve the Time-dependent Schrödinger equation. One of JamalS' recommendations was the Fourier transform, which is why I want to quote ...
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245 views

Time-dependent Schrödinger equation with $V=V(x,t)$

I was wondering about the following: If you have the time-dependent Schrödinger equation such that $$i \hbar \frac{\partial\psi(x,t)}{\partial t} = - \frac{\hbar^2}{2m} ...
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1answer
86 views

How is the time independent potential term a solution of Schrodinger equation

Consider a time-independent potential: $V(x)$. Then, it is usually stated that $$ \Psi(x,t)=\rho(x)\exp{\left(-\frac{i}{\hbar}Et\right)} $$ is the general form of a solution of the Schrodinger ...
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Calculating Quantum number from initial conditions

I have solved the particle in a box problem to get energy eigenstates and wave vectors: $$E_{n}=\frac{\hbar^{2} k^2}{2m} ,\hspace{1cm} k_{n}=\frac{\pi n}{L}$$ And now I am trying to figure out how ...
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Particle in a box under harmonic driving

Is the particle in a box under harmonic driving electric field solvable analytically? Here is the Schrodinger equation: $$ i\frac{\partial \psi(x,t)}{\partial t}=\left[-\frac{1}{2} ...
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1answer
79 views

Quick question on perturbation theory

Suppose we have a particle in an infinite potential well, with $V(x) = 0,\space 0< x < a $ and infinity everywhere else. Now suppose we have a perturbation on the LHS of the well: $V_1(x) = v, ...
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55 views

Stabilization of von Neumann equation

Given the solution of the Von Neumann equation $\rho(t) = e^{-i H t/\hbar} \rho(0) e^{i H t/\hbar}$ How can we justify if it will be stabilized as $t\rightarrow\infty$ in general? For example, ...
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44 views

Time evolution of states - Is total energy constant or not?

Suppose the state of the particle is given as follows: $$ |\psi_{(t)}\rangle = \frac{1}{\sqrt2} \left( e^{-\frac{i\omega t}{2}} |0\rangle + e^{-\frac{3i\omega t}{2}} |1\rangle \right) $$ Where the ...
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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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136 views

Why are hydrogen energy levels degenerate in $\ell$ and $m$?

Is there a good physical picture of why the energy levels in a hydrogen atom are independent of the angular momentum quantum number $\ell$ and $m$?
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1answer
39 views

Nuclear shell model - finite square well

I am trying to make a simplified approximation and solve Schrodinger equation in the finite square well to model the nucleus of Ca (shell nuclear model). The potential is $ V(r) = -V_0$ for ...
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2answers
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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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1answer
64 views

Time-dependent perturbation - details in derivation

I get confused about two things when deriving the time-dependent perturbative approach. We have the Hamiltonian $$H = H_0 + \lambda H^{(1)}$$ and we have solved (from Schroedinger) $$\dot{C_f(t)} ...
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1answer
56 views

Two explanations of non-zero atomic radius

I have came across two separate explanations for why atoms have a positive atomic radius (as opposed to electrons "collapsing" into the nucleus). The first is via Heisenberg Uncertainty Principle, ...
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150 views

Isn't $E-U = K$ in Schrodinger's Equation?

I'm studying quantum mechanics in its most basic level (I don't even know if Physicists call this already quantum mechanics) and I have one doubt in Schrodinger's equation. The book presents the ...
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104 views

Numerical solving of 2D and 3D Schrodinger equations

I am studying 2D quantum scattering models for my Bachelor's thesis. Somewhat like these: ,with Dirichlet ($\psi \mid_\Gamma = 0$) boundary conditions on the "walls" of the waveguide and the ...
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1answer
168 views

Time derivative of the state vector as expressed in abstract Hilbert space vs. as a wavefunction

The Schrodinger equation in Hilbert space is expressed as : $$\frac{\partial}{\partial t} \psi(t) = \frac{-i}{\hbar}H\psi(t). $$ Here $\frac{\partial}{\partial t} \psi(t) \equiv \psi'(t) \equiv\lim ...
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2answers
85 views

The nature of “hard wall” boundary condition for Schrodinger's equation

For a quantum particle in an one-dimensional infinite well of width $L$, the potential has the formal expression: $$ V(x) = \begin{cases} \infty, & x < 0 \\ 0, & 0 \le x \le L \\ \infty, ...
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3answers
660 views

An operator on the other side of the Schrödinger equation

A form of the Schrödinger equation is $$ \left[-\frac{\hbar^2}{2m} \nabla^2 + V(\vec{r}, t)\right]\Psi = i\hbar \frac{\partial}{\partial t} \Psi $$ The bracketed term is of course the ...
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1answer
66 views

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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84 views

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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1answer
74 views

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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1answer
70 views

Question about the Klein Paradox

I am trying to understand some points about the paradox, what I am doing is solving the for the step potential $$ V = V_0 ~\theta(z) $$ I have two solutions $$ \Phi_I = e^{ik_1 z} + r e^{-ik_1 z}$$ ...
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2answers
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Can the wave function of the hydrogen atom be written as a superposition of plane waves with a single frequency?

The solutions of the Schrödinger equation for a hydrogen atom have definite energy. Does this mean that they could be written as a superposition of plane waves of a single frequency - corresponding to ...
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1answer
50 views

Extending a solution for $\left(H_\text{osc} + \delta^{(3)}(\vec r) \frac{\partial}{\partial r} r \right) \Psi(\vec r)$ to Gaussian potentials

I am a undergrad and currently trying to understand and use the following paper: Th. Busch u. a. Two Cold Atoms in a Harmonic Trap. 1997. URL: ...
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211 views

Boundary conditions: Particle in a ring

Particle in ring is a well-known example where a solution of the Schrodinger equation exists. My question is: In principle we also want that $\psi'(\theta) = \psi'(\theta + 2\pi)$. The thing is that ...
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1answer
65 views

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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1answer
77 views

Infinite potential well with barrier in the middle- symmetric

So I'm having problems with the double infinite potential well given by $$V(x)= \left\{\begin{array}{ll} \infty & -\infty < x < -a-b \\ 0 & -a-b< x < -a \\ V_0 & -a < x ...
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1answer
94 views

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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2answers
169 views

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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2answers
34 views

Schroedinger equation. Mass. Charge

The mass of a particle appears in Schroedinger equation but it does not appear its charge, although both terms have their effect on movement.Why?
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2answers
95 views

Schroedinger equation. Why Potential energy instead of Force?

What is the reason Schroedinger equation is quoted in terms of potential energy instead of force?
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72 views

Finding the odd parity bound state wave function for a particle in one dimension

Let a particle of mass $m$ and energy $E$ be moving on the $x$-axis in a potential $V(x)$ given by $$V(x)=\left\{\begin{matrix} -V_{0}, & -a<x<a\\ 0, & otherwise \end{matrix}\right.$$ ...
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1answer
69 views

Change of variable in harmonic oscillator time independent Schrodinger equation

I was revising the harmonic oscillator for my intro to quantum course and realised I'd sort of accepted a change of variable result without actually being able to get to it. It says: The stationary ...
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Solving the Schrodinger equation with appropriate symmetry

In the paper Markov Fields by Edward Nelson the introduction section claims that analytically continuing a Markov process with appropriate symmetry properties yields the solution of the Schrodinger ...
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2answers
43 views

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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How does the hydrogen atom know which frequencies it can emit photons at?

At university, I was shown the Schrodinger Equation, and how to solve it, including in the $1/r$ potential, modelling the hydrogen atom. And it was then asserted that the differences between the ...
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1answer
62 views

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. ...