Questions tagged [schroedinger-equation]

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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How do I use first-order perturbation theory to compute the first four energy levels related to a potential well?

I came across a problem whose statement is as follows: An electron moves in the potential well $P (x) = -\delta$ for $- a <x <0$ and $P (x) = \delta$ for $0 <x <a$ (Fig. 13.7). Use first-...
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How to correct eigenvalues for infinite square well potential using numerov matrix method? [closed]

i was using this code infinite was approximated by 10^10. But the energy eigen values that i got is completely different from the actual .How to resolve ...
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Solving the energy spectrum for a $\cot^2$ potential well using the WKB approximation

I am taking graduate quantum mechanics, and we are now discussing the WKB/semiclassical approximation. I am trying to solve a problem where I would need to find the energy spectrum of the 1D motion in ...
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Can we handle the wave function as if it was a real valued function? [duplicate]

I am trying to analyze in general simple one dimensional QM problems. To be more specific let's consider this kind of Hamiltonian: $$H=\frac{\hat{p}^2}{2m}+V(\hat{x})$$ From this one we can derive the ...
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3d solutions for 1d Schrödinger equation?

The general Schrödinger equation in 3d is $$i\hbar\frac{\partial\psi}{\partial t}(\mathbf r, t)=-\frac{\hbar^2}{2m}\nabla^2\psi(\mathbf r, t)+V(\mathbf r)\psi(\mathbf r, t).$$ Now consider that $$V(x, ...
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How to put laser field into the Hamiltonian of the Schrodinger equation of a 3-level quantum system?

I read a paper about using femtosecond laser to control a 3-level quantum system. The author wrote the Schrodinger equation for the system and wrote the expression of the laser field. But I still don'...
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Separation into $|\Psi_{CM}\rangle$ and $|\Psi_{R}\rangle$ in abstract representation?

Consider a two-particle system with Hamiltonian $$\hat H = \frac{(\hat{\mathbf p}^{(1)})^2}{2m_1} + \frac{(\hat{\mathbf p}^{(2)})^2}{2m_2} + V(\hat{\mathbf r}),$$ where $\hat{\mathbf r} = \hat{\mathbf ...
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Direct Series Solution Attempt of the Quantum Harmonic Oscillator

The non relativistic Schrodinger equation of the harmonic oscillator in dimensionless variables is $$\frac{d^2 \Psi}{d \xi^2} = (\xi^2 - k)\Psi$$ where $$k \equiv \frac{2E}{\hbar \omega}$$ According ...
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Can two Hamiltonians be added using the Heaviside function?

Consider two Hamiltonians $\hat{H}_1(x)$ and $\hat{H}_2(x)$ that are defined for negative and positive values of x respectively, such that I can combine these two Hamiltonians to one: $$\hat{H}(x) = \...
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Expectation value in the Schrödinger Equation

The Schrödinger equation is given by: $$ i \hbar \partial_t |\Psi\rangle = \hat{H}|\Psi\rangle $$ The right hand side is just an operator acting on a state vector, so we are free to consider its ...
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Does the time-independent Schrodinger equation in 1D have an exact and general solution?

The (time-independent) Schrödinger equation is for sure the most important equation in quantum mechanics: $$-\frac{\hbar^2}{2m}\nabla^{2}\psi(\vec{r}\,)+V(\vec{r}\,)\psi(\vec{r}\,)=E\,\psi(\vec{r}\,).$...
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Transition between 2 energy levels - wave function picture

Suppose we have a system that has discrete energy levels (e.g. hydrogen atom, potential well) and the stationary solutions for the wave function are $\psi_n$. I would assume that there should be a way ...
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Do quantum wave functions rotate through imaginary space?

Watching a visualization of Schrödinger’s equation, I noticed that the wave function for a 2-dimensional particle was placed in a 3-dimensional graph consisting of 2 Real axes and an Imaginary axis. ...
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Why is Schrodinger equation taught while it does not describe an electron?

Strictly speaking, it is "wrong" because it does not describe spin-1/2 particle like an electrons. Why in every QM textbook is it taught, not as a historical equation, but as a current ...
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Radial Schrödinger equation of a scattering in two dimentions

The scattering, in two dimensions, of a particle of mass $m$ by a central potential $U(r)$. The hamiltonian of the system is $H= p^2/2m + U(r)$. Then the radial wave function $ϕ(r)$ is obtained as a ...
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Treating the delta potential in a Schroedinger equation in 1D

It is a standard problem in quantum mechanics. For the equation $$ -\psi'' + g \delta(x) \psi = E \psi ,$$ we integrate from $-\epsilon$ to $+\epsilon$ and thus get the boundary condition $$ g \psi(0) ...
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How could wavefunction be the same and yet it has degenerate eigenvalues?

In solid state physics, the schrodinger equation $$H \psi_{\vec{k}} = E_{\vec{k}} \psi_{\vec{k}}$$ has solutions $\psi_{\vec{k}}(x)$. In the near free electron approximation, I was told that $$\psi_{\...
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Finite double potential barrier transmission coefficient

TL;DR: I want to calculate the transmission coefficient of a particle travelling into a finite double potential barrier system and I think I've got stuck by the fact that I have 9 unknown variables (...
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Solve analytically the time-independent Schrödinger equation for the Hamiltonian $\hat H$

The Hamiltonian $\hat H$ is given by: $$\hat H=\frac{1}{2}(\hat p^2+\hat q^2)+K(\hat p\hat q+\hat q\hat p)$$ where $K$ is a real constant, in the coordinate space: $$\hat p=-iℏ\frac{\partial}{\partial ...
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How can matter, the things which we can touch and feel on a macro level, exhibit a wave nature?

Louis de Broglie suggested that, if a particle like electron has momentum and wavelength associated with it (due to Planck's constant), then it might be a wave. The region where it exists are those ...
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Variational method: Why do parameters differ for two trial functions (optimization)?

Below the potential and trial functions: $$V(x)=(x^2-1)^2-x^2$$ Use the variational method with the two trial wave functions: $$\psi_{\pm}(x)=A\left(e^{-\frac{(x-x_0)^2}{2\sigma^2}}\pm e^{-\frac{(x+...
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Solving for the momentum from eigenfunctions

When you have a solution to a time-dependent Schrodinger Equation, $$\Psi(x,t)=\exp\left({-\frac{i\hbar^2k_0^2t}{2m}}\right)\sin(k_0x), \tag{1}$$ and want to know the distribution of momentum at time ...
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Adding λI to the Hamiltonian has no impact? [closed]

Show that If we add λΙ in H, where I is the identical operator and $λ\in\mathbb{R}$, it won't affect any measurement.
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Are solutions to the particle in a (finite) box problem orthonormal?

Solving the particle in a box problem is fairly straight-forward for both, the finite and infinite potential well. While it is well known that the solutions to the infinite potential well must be ...
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For a radially symmetric wavefunction is $\Psi$ allowed to blow up at $r=0$ provided that $|\Psi|^2r^2$ doesn't?

For a spherically symmetric wavefunction the probability is proportional to $|\Psi|^2r^2$, and if the wave function blows up at $r=0$ then at $r=0$ $|\Psi|^2=\infty$, and $r^2=0$ meaning that the ...
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Why do different energy levels affect the probability amplitude of the wave function?

When you solve the Schrödinger equation for the classic particle in a box you get that $$\psi=\sqrt{ \frac{2}{l}} \sin{(\frac {n\pi x}l)}$$ where $x$ is the length from the leftmost point of the box, $...
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Solutions of the Harmonic Oscillator are *not* always a Combination of Separable Solutions?

Are there solutions of the Schrödinger equation that are not a linear combination of separable solutions and how do we find them? In Griffiths, Quantum, Prob. 2.49, there is a solution of the (time-...
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On the 1D Quantum Mechanics Harmonic Oscillator

I was solving the P. 2.41 of Griffiths' Introduction to Quantum Mechanics. Nothing really new until I read a proposed solution (from Griffiths' himself) for the problem in which it states that I can ...
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What is the Schrödinger equation used for exactly?

The Schrödinger equation is just another way of writing the conservation of energy, right? So how can you use it to find the quantum wavefunction? I mean in every example I've seen the wavefunction is ...
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Normalising a wave-function

So I have a small confusion when normalising an infinite well wave-function. The wave-function for my problem is $$Ψ(x) = Ae^{i(kx-wt)}+Be^{-i(kx-wt)}+Ce^{i(kx-wt)}+De^{-i(kx-wt)}.\tag{1}$$ Applying ...
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Deriving the operator version of the classical wave equation

From the book i studying from to 'derive' the Schrodinger equation this is part of the process: Two differential operators, $\hat{p}$ and $\hat{E}$, representing the classical momentum and energy ...
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Why does Hartree-Fock work so well?

Why does the Hartree-Fock method for electronic structure work so well for atoms? More specifically, why is the "correlation energy" a relatively small component of an atom's (ground state) ...
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Why does the Schrodinger equation depend on conservation of energy if conservation of energy is violated

The Schrodinger equation: -((h^2)/8 pi m)(d^(2)psi(x)/dx^2)+v(x)psi(x)=E psi(x) is just another way of writing : kinetic energy + potential energy = total energy right? (for those of who can't see ...
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Time evolution of the Gaussian packet

I am trying to get the time evolution for the following initial condition: $$ \Psi(x,0) = \left(\frac{1}{2\pi \sigma^2} \right)^{\frac{1}{4}} e^{- \left(\frac{ x-x_{0}}{2 \sigma}\right)^{2}} e^{i\frac{...
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What does $V(r)$ mean in the Schrodinger equation?

The Schrodinger equation: $$-\frac{\hbar^2}{2m}\nabla^2\Psi(r)+V(r)\Psi(r)=E\Psi(r)$$ $$\textit{kinetic energy} + \textit{potential energy}=\textit{total energy}$$ Is one of my favourite equations, ...
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B.C. for time-decaying delta barrier inside an infinite well

So let's say I have an infinite well with walls up at $x=-L$ and $x=L$. Suppose that inside the well, there is a time-dependent potential $$ V(x,t)= \alpha_0\delta{(x)}f(t) $$ where $f(t)$ is a ...
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Do energy levels ever appear at the peak of a cosine potential in higher dimensions?

I simulated a hyperspherically symmetric wavefunction for the case of $V=-\cos(r)$, in which $h=1$, and $m=1$ and the number of spatial dimensions is $4$. I charted the integral of the square of the ...
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How can we form Euler-Lagrange equations for Time-Independent Schrodinger Equations?

Is it possible to form a lagrangian of the TISE using the concept of Lagrange Multipliers? I am new to this topic so any help would be much appreciated.
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Finding the wave function at all times for a particle trapped in an infinite potential well, given the initial wave function of the particle [closed]

A particle is trapped in the region $0 \leq x \leq \pi$ by an infinite potential well. At the initial time, the particle has a wavefunction $\psi(x, 0)=A(\sin (x)+\sin (2 x))$. Find the wavefunction ...
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Determining the motion of a classical particle and quantum particle in a given potential with a given initial position and energy [closed]

How do I go about solving this? I know bound states require the energy to be less than the potential while scattering states require the energy to be greater than the potential. I also know that ...
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How does a sine wave evolve under the Schrodinger equation?

On pg 72 of "Something Deeply Hidden," Sean Carroll discusses how the uncertainty principle is just a consequence of the Schrodinger equation. He writes: Consider a simple sine wave, ...
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Restrictions on Initial Values for the first derivatives of a wavefunction, for a bound state in the time independent Schrödinger Equation?

The time independent wave function for a bound state given some potential function $V(r)$ is given by the time independent Schrödinger Equation $$E\Psi=-\frac{\hbar^2}{2m}\left(\frac{\partial^2\Psi}{\...
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Solving Schrodinger equation for the hydrogen atom

In University Physics with Modern Physics by Hugh Freedman in Chapter 41.3 they go about solving the Schrodinger equation for the hydrogen atom. At one step they say to substitute the following ...
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Compute the value of $\lim_{x \to-\infty}Ae^{ikx}+Be^{-ikx}$ [closed]

(After some recommendations , I feel I need to elaborate a little bit more on my question) So I was solving Schrodiger's equation for a step: $$ V(x)= \begin{cases} 0 & x<0 \\ V_0 & x>...
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How is retarded Green's function related to 1D scattering coefficient?

I have been reading a paper about Green's function with 1d potential barrier https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.114.126602 Supplementary material Sec. II $H=\hbar^2k^2/2m+V\...
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Diagonalization of Time dependent Hamiltonian using ZHEEVR

I need to calculate all the eigen values of the time dependent Hamiltonian using ZHEEVR, but I don't know how to define the time dependent Hamiltonian matrix. Please help in this regard
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Understanding the Schrodinger's equation by variational principle

I reviewed part of my notes in the quantum mechanics class, and still have a few questions about the variational derivation of the Schrodinger's equation: The variational principle says that the ...
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Continuous non-relativistic bound states

Consider a group of charged point(at least considered as such in this non-relativistic limit) particles such electrons,protons , nucleii alone in an empty infinite universe and NOT considering any ...
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How do you use the Schrödinger Equation to find $E_n$ as a function of $n$ for any potential function $V(r)$?

I understand that for the hydrogen atom $$V=-\frac{e^2}{4{\pi}{\epsilon_0}r}$$ and $$E_n=-\frac{m_e^2e^4}{8\epsilon_0^2h^2n^2}$$ In the case of a parabolic potential $$V=\frac{1}{2}Kr^2$$ and $$E_n=\...
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What happens if you, an energy that's between two energy levels for bound states, into the Schrödinger Equation? [closed]

As I understand it in Non Relativistic Quantum Mechanics bound states are only allowed at certain energy levels, and that this is the case for any potential that has bound states. Also for some ...

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