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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Particle in a 2D box degeneracy [on hold]

I'm doing revision for my quantum test next week and my university, irritatingly, doesn't provide mark schemes to past questions. I feel like I've done most of this one right, but the end has thrown ...
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1answer
30 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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77 views

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
35 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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39 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. ...
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2answers
103 views

Computer simulation of Schrödinger equation [duplicate]

I am looking for a computer program which simulates the Schrödinger equation (say for a single particle) in two dimensions and for potentials and initial states specified by the user. Typical ...
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1answer
44 views

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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58 views

Free-particle solution to Schrödinger Equation

The free particle solution in stationary state (with definite energy) to the Schrödinger equation is $$\psi(x,t) =Ae^{i(kx-\omega t)} + Be^{-i(kx+\omega t)}$$ Since the energy is definite, and ...
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1answer
45 views

Solving the 1-d time-independent Schroedinger's equation with an infinite boundary

In my introductory modern physics class we have examined time-independent solutions to the Schrödinger equation in 1 dimension. We looked at a few cases without finite boundary, e.g., free particles ...
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1answer
34 views

Reflection of an evanescent matter wave within a finite barrier?

To my understanding if I have a finite barrier with potential $V(x)>E$, then to the left of the barrier, the wavefunction can be represented as two exponentials: $$\psi= e^{(ik_{left} x)} + ...
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138 views

Lie group of Schrodinger Wave equation

In Ballentine's book on quantum mechanics (in 3rd chapter), he introduces the symmetry transformation of Galilean group associated with Schrodinger equation. Now the Galilean group as such has 10 ...
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1answer
149 views

Harmonic Oscillator potential, proof that Gaussians remain Gaussians?

I read in several papers that for a Harmonic Oscillator Hamiltonian in the time dependent Schrödinger equation a Gaussian wave packet remains Gaussian. Unfortunately I could not find any proof for ...
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173 views

The formal solution of the Schrodinger equation

Let's have Schrodinger equation (or some equation in Schrodinger form) $$ \tag 1 i \partial_{0} \Psi ~=~ \hat{H} \Psi . $$ One likes to write that it has formal solution $$ \tag 2 \Psi (t) ~=~ ...
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2answers
52 views

Schrödinger's Equation and its complex conjugate

I would like to know why there is a minus sign on the right-hand side of the Schrödinger's complex conjugate equation, whereas in the Schrödinger's equation there isn't. I know it is a simple ...
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1answer
40 views

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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4answers
317 views

How to derive Schrödinger equation?

How is the Schrödinger equation $$\frac {\partial }{\partial t}\psi=-\frac {i }{\hbar }H{\psi }$$ being derived?
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52 views

Interesting Harmonic Oscillator Solution

On page 89 of Griffith's QM book, an exact solution to the time-dependent SE equation for the harmonic oscillator is mentioned: $$ ...
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1answer
48 views

Relating Schrödinger's Wave Equation and Heisenberg Uncertainty Principle

A homework question that I don't conceptually understand: A quantum particle of mass M is trapped inside an infinite, one-dimensional square well of width $L$. If we were to solve Schrodinger's wave ...
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222 views

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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2answers
73 views

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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2answers
348 views

Infinite and Finite Square Wells

For the infinite square well in the first region, outside the well: $$\frac{-\hbar^2}{2m}\frac{d^2 \psi}{dx^2} + V(x) \psi (x) = E \psi (x),$$ where you set $V = 0$. Rearranging gives $$\frac{d^2 ...
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2answers
123 views

Eigenvalue problem for differential equations in QM

I have a very simple question with regard to numerical methods in physics. I want to solve the eigenvalue problem for a particle moving in an arbitrary potential. Let's take 1D to be concrete. I.e. I ...
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1answer
53 views

Is there a closed form expression for Landau-level eigenstates?

Is there a closed form expression for the Landau-level eigenstates (preferably in the symmetric gauge)? This is the 2-dimensional quantum mechanical problem of a charged particle moving in a ...
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1answer
69 views

weak solution of Schroedinger equation .. are they useful?

We know that Schroedinger equation can be deduced from a variational principle (non relativistic Schroedinger equation). Assuming this I have 2 questions: a) Using variational methods, could we ...
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52 views

Casimir operator for the $sl(2)$ group [duplicate]

The generators of the $sl(2)$ group can be expressed as: $$J^+ =z\frac{d}{dz}-2jz, \quad J^0=z\frac{d}{dz}-j, \quad J^-=\frac{d}{dz}$$ How would find the Casimir operator for such a group?
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Finding the energy eigenvalues of Hydrogen using WKB approach

I need help to find the energy eigen values of Hydrogen atom using WKB approach. So far I know, the radial equation is given by $$\frac{1}{r^2} \frac{\partial }{\partial r} \left( r^2 \frac{\partial ...
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Particle in a higher-dimensional box with an attractive delta potential

Suppose you have a particle in the box $[0,L]^d$, with an attractive Dirac delta potential $-\delta_{\vec w}(x)$ at $\vec w$. How do you solve the Schroedinger equation for this system? In the case ...
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58 views

Philosophical take on Schrodinger's cat paradox

I'm a mathematics undergrad currently doing a module on quantum mechanics. I am interested to delve into the more philosophical aspect of the paradox. Could anyone please give me some introductory ...
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55 views

Understanding the algebra associated with an implicit potential

In the paper here(page 7-8) the authors make a claim that the Natanzon potential (an implicit potential) follows an $SO(2,2)$ algebra. This potential defined as : $$ U(z(r)) = ...
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2answers
129 views

Harmonic Oscillator - Energy quantisation

The one-dimensional quantum HO can be solved in Schrodinger representation by getting Hermite Differential Equation $$ \frac{d^2y}{dx^2} - 2x \frac{dy}{dx} + \lambda y = 0 $$ with solutions $$ y(x) = ...
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1answer
78 views

Applying the Schroedinger equation

I hope this isn't a dumb question, but... If we have two fixed sine waves, both of which have a frequency range of +1 to -1, with a ratio between the waves of wave(1):3 to wave(2): 1, what does the ...
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98 views

Determining the group associated with a given potential?

I'm trying to understand how symmetry groups are related to potentials of the Schrodinger equation. In particular, I wish to know if it is possible to find the symmetry group of this potential $$V(x) ...
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88 views

Infinite well particle subject to additional time dep. potential

I am asked to find the wavefunction of the particle in a well subject to an additional potential $$V(x,t)=\frac{\pi x \hbar}{L}\delta(t).$$ I have already solved that ...
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151 views

How do electrons and photons interact?

Two electrons, or an electron and a proton, interact with each other because of the Coulomb potential, which can also be seen in the Schrödinger equation (which is the equation that describes the ...
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195 views

Non-separable solution for the Schrödinger equation

Schrödinger solutions are usually if not always of the type: $\psi=\operatorname{T}(t)*\operatorname{X}(x)$ (we use the separation of variables method to arrive at the time independent Schroedinger ...
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71 views

Fixed Angular Momentum

Say I'm given the following Schrodinger equation $$\frac{d^2u}{dx^2}+ \left[E - V(x)+ \frac{a}{x^2}\right]u(x) =0$$ Where $a \in \mathbb{R}$. What are the physical interpretations of this equation? ...
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1answer
83 views

Differential Realizations of certain algebras

I'm a first year graduate student in Mathematical Physics, and I am trying to generalise a certain method involving the so-called "Differential realizations" of certain algebras. The problem I'm ...
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3answers
166 views

Is any time-dependent hydrogen atom Schrödinger equation solvable analytically?

It's well-known that hydrogen atom described by time-independent Schrödinger equation (neglecting any relativistic effects) is completely solvable analytically. But are any initial value problems for ...
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167 views

Proof of the time-independent Schrödinger equation

I have a question regarding the proof of the time-independent Schrödinger equation. So if we have a time-Independent Hamiltonian, we can solve the Schrödinger equation by adopting separation of ...
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1answer
99 views

TISE for a triangular potential

I want to solve the TISE of a particle of charge $q$ and mass $m$ in a one dimensional triangular potential, with an infinitely high potential wall at $x = 0$, i.e. ...
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2answers
184 views

Solving quantum radial equation for infinite potential spherical annulus for $l=0$

There is a mass $m$ in a potential such that $$ V(r) = \left\{ \begin{array}{lr} 0, & a \leq r \leq b\\ \infty, & \text{everywhere else} \end{array} \right. $$ ...
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1answer
104 views

Change of operator in the Hamiltonian [closed]

We are told that the particle has mass m and charge e and is moving in 2 dimensions. The position operator $\mathbf{X}=(X_{1},X_{2})$ and momentum operator $\mathbf{P}=(P_{1},P_{2})$ We are given ...
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1answer
136 views

Virial theorem and variational method: an exercise (re-edited)

I have a hydrogen atom, knowing that its Hamiltonian has been modified turning the standard potential $$ V_{0}(r) = -\frac{Z}{r} $$ into $$ V(r) = -\frac{g}{r^{\frac{3}{2}}} $$ with $g$ a positive ...
2
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1answer
81 views

Does this commutation relation hold?

I was wondering whether it is true that $[L_x^2,x^2+y^2+z^2]=0$. I could not find it in the internet and therefore I wanted to ask here whether anybody here knows that this is true or false.
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1answer
91 views

Infinite potential square well solutions

My question is about understanding the different solutions of the potential square well. Imagine a square well defined this way: $$ V(x) = \begin{cases} ∞&\,{\rm if} x<0 \\ 0&\,{\rm ...
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2answers
156 views

Wave function with a delta potential

I have a particle and a potential $V(x)=\frac{\hbar^2}{2m}k\delta(x)$. Where $\delta (x)$ is the Delta function, and I am interested in the solutions of the stationary Schroedinger equation. If ...
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2answers
231 views

Harmonic oscillator modified by infinite well: are analytic solutions possible?

I'm trying to find solutions to a harmonic oscillator that sits within an infinite square well. I haven't spent too much time yet, and I've had no success so far. I'm wondering how possible or complex ...
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3answers
611 views

Why is a proton assumed to be always at the center while applying the Schrödinger equation?

Why is a proton assumed to be always at the center while applying the Schrödinger equation? Isn't it a quantum particle?
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179 views

Can we have a physical interpretation for a time independent Schrodinger equation of this form?

I am interested in a time independent Schrodinger equation of this form. $$F*\psi - \frac{\hbar^2}{2m} \frac{\partial^2{\psi}}{\partial{x^2}} = E\psi$$ Here the product $V\psi$ is replaced by the ...