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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Infinite 3D sphere well with Dirac Delta potential function at the origin

A spinless particle of mass $m$ is constrained in a 3D region of zero potential within an impenetrable spherical shell of inner radius $r = a$, with a delta function potential at the origin given that ...
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How does the wave function relate to probability?

I'm trying to solve this problem which involves the probability of a particle being in a certain region. I know that $|\Psi|^2$ is the probability density, but how do I get this in the region?
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Schrodinger equation with parameters

I need to know the ground state energy $E_0$ defined by the following stationary Schrodinger equation: $$ -\frac{a}{2}\phi''(\xi) + \left(\frac1{2a}\sinh^2(2\xi) + (2b-1)\cosh(2\xi)\right)\phi(\xi) = ...
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What is the impact of the time-independent Hamiltonian operator on the observation probabilities?

If you assume that $H$ is a time-independent Hamiltonian, by the Schrodinger equation, the state evolution $|\Psi(t)\rangle$ is given by $ \left( \sum e^{\frac{-i \lambda_j}{\hslash} t} |{ v_j }\...
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Doubt in the completeness of wave function

I am reading about the completeness property of wave function. The following is given about it- The energy eigenstates are complete in the sense that any reasonable wave function $\psi(x)$ can be ...
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QM - Step Potential - Comparing Reflection Probabilities [closed]

I am currently studying, and have ran into an issue concerning a problem. It uses a particle coming in from the left and a step potential of $U_0$ at $x = 0$. The energy is always positive but in this ...
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Predicting the probability distribution in a potential

I've been dealing with a kind of problem in quantum mechanics, where they give us an arbitrary potential, and then ask us to predict the form of the probability amplitude or the wave function. The ...
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What is the general solution of one-dimentional time-independent Schrodinger's equation?

As I tried to learn quantum mechanics I have found two solutions of one-dimensional time-independent schrodinger equation in various resources. One is,$$\psi(x) = Asin(kx)+Bcos(kx)\\\text{where}, k = \...
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How to reconcile two different derivations of the time-independent Schrödinger equation?

On one hand, using the Spectral decomposition of the Hamiltonian operator $H$, assumed to be an Hermitian operator, it is relatively simple to derive the equation $U(t) = \sum |v_j\rangle\langle v_j| ...
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Schroedinger cat states of the harmonic oscillator

I've found in an article that it is possible to prepare experimentally the superposition of two coherent (quasi-classical) states to obtain the Schroedinger cat state: $$ \left|\psi_{\pm}(t)\right\...
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Assumption made for the WKB approximation in radial coordinates

I was thinking the other day, if you had the Schrodinger equation in 3-dimensions, and had a spherically symmetrical potential. Ie.: $$-\frac{ℏ^{2}}{2m}∇^{2}ψ+V(r)ψ=Eψ$$ Then you could simplify the ...
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How can Bound state energy be negative if the $V_{min}$ is positive?

We know that Energy must be negative for bound states (as the wavefunction must go to 0 at infinity) but when we are looking at potential wells, we also say that E must be greater than the minimum ...
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Born Rule proof for freshmen

Although early quantum mechanics are taught in many freshman courses, the Born Rule is almost never proved at that stage. Is it even impossible to elementarily prove that the probability density is ...
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Solving the Schroedinger equation with the initial condition as an energy eigenstate [closed]

I was studying quantum mechanics by watching a video lecture series. In the lecture https://youtu.be/TWpyhsPAK14?list=PLUl4u3cNGP61-9PEhRognw5vryrSEVLPr&t=2784 , the professor tries to solve the ...
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Expectation value of $\frac{1}{r^2}$ of hydrogen atom [duplicate]

I'm studying on the derivation of $\left\langle \frac{1}{r^2} \right\rangle$ by using the book Nouredine Zettili. The derivation in the book is as follow: I can understand the rest of the derivation ...
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Why is the reflection coefficient 1 for step potentials where energy is less than the potential?

Consider a potential $V(x)$ which is zero when $x<0$ and $V_0>0$ when $x>0$. Suppose there is an incident particle with momentum $p=\hbar k$ and energy $E = \hbar^2 k^2 / 2m < V_0$ coming ...
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I am having a doubt in graphs of $4πr^2|\psi|^2$ vs $r$ and $4πr^2|R(r)|^2$ vs $r$

To show radial probability, in some sources they used the graph $4πr^2|\psi|^2$ vs $r$ In some other sources, they used the graph $4πr^2|R(r)|^2$ vs $r$ However, $\psi(r,\theta,\phi)=R(r)Y(\theta,\phi)...
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Why is the Time Independent Schrodinger Equation so important? [closed]

The main equation of Quantum Mechanics (QM) is the Schrodinger Equation (SE): $$i\hbar\frac{\partial \psi (x,t)}{\partial t}=H(x,t)\psi(x,t)$$ Why is this equation so important? It's important because ...
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Negative Potential Step - What happens when it isn't sharp (there is a width $a$)?

I'm wondering what happens when you take a normal negative potential step, but then give it a width $a$ instead of a straight drop in potential. As the width, $a$, gets bigger, what would happen to ...
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Wave Function for a Step Potential

If we sole the TISWE, and if energy or the particle lies between 0<E<V. If we do the calculation, Transmission coefficient (T) comes out to be zero. I get that part, but why then there exist a ...
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Is it true that Schrödinger wrote his Schrödinger Wave Equation from his mind? [duplicate]

My physics teachers told me that there is no derivation of the Schrödinger Wave Equation and that Schrödinger actually wrote this equation from nothing. He wrote it from his mind. Even in my book ...
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How to derive the effective Hamiltonian in CP violation?

https://www.nikhef.nl/~h71/Lectures/2015/ppII-cpviolation-29012015.pdf This note (and many other notes or textbooks) on CP violation introduces the SM Lagrangian $L_{SM}=L_{kinetic}+L_{Higgs}+L_{...
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In quantum mechanics when we use the real wavefunctions to find the average value of momentum operator then it comes out be zero. What does it mean? [closed]

In quantum mechanics when we use the real wavefunctions to find the average value of momentum operator then it comes out be zero. What does it signifies? Please explain it.
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Scattering from finite square well and the transmission coefficient

Suppose we have a typical finite square well where $\lim_{x \to \pm\infty} V(x)=0$ and $V(x)=-V_0$ for all $x\in[-a,a]$ where $V_0>0$. The finite square well admits both bounds state solutions (...
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For the even wavefunction and odd wavefunction, can we estimate whether the energy of the system is positive or negative?

For the even wavefunction and odd wavefunction, can we estimate whether the energy of the system is positive or negative? And for which of (odd or even wavefunction) energy is higher? You can consider ...
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Is there any practical potential for which first derivative of wavefunction is continuous? [duplicate]

As we know that first derivative of the wavefunction is discontinuous when the potential is infinity. Is there any practical potential for which first derivative of wavefunction is continuous?
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How to solve the Schrödinger equation for the hydrogen atom?

We know that the time-independent Schrödinger equation, if we are using Cartesian Coordinates, is given by: $\begin{aligned}-\frac{\hbar^{2}}{2 m}\left(\frac{\partial^{2} \psi(x, y, z)}{\partial x^{2}}...
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$k$-dependence of the energy in solid state physics

In a crystal, the electrons are subject to a periodic potential due to the fact that the atoms form a periodic lattice. From this periodicity we can obtain the Bloch theorem, and get a general formula ...
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On using Python to solve Time Independent Schrodinger Equation, the eigenfunctions have their values “pushed” to one of the boundaries?

I am having trouble using numerical methods to solve Time Independent Schrodinger Equation. I am considering a quartic potential function: $$ V(x) = x^4 -4x^2.$$ $$ -\frac{d^2\psi(x)}{dx^2} + V(x) \...
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Doubt during derivation of Ehrenfest's theorem [duplicate]

I was reading about the derivation of Ehrenfest's theorem in this website when I came across this step: Substituting from Schrödinger's equation (137) and simplifying, we obtain $$\frac{d\langle p\...
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188 views

Probability density of a free particle

I have been recently studying QM and I have encountered the case of a free particle. I understood that a free particle travels in the form of a wave packet where we get $$\psi (x) = \frac{\int_{-\...
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Determining coefficients for wave function solutions of an electron in a periodic potential

In Kittel's Intro to solid state physics, when solving the schrodinger equation for a periodic potential, we begin by writing the potential and the wave function as fourier series of the form $$\psi = ...
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Finite square well and continuity [duplicate]

In solving finite square well problem, we solve the TISE inside and outside the well, and we match the wave function at the boundary, by the continuity of wave function. Now this bugs me, since the ...
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Schrodingers equation for an electron in a periodic potential derivation

In Kittels Introduction to Solid State physics, when deriving schrodingers equation for an electron in a periodic potential, we begin by writing the wave function as a Fourier series $\psi = \sum_k C(...
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Solving the hydrogen atom in parabolic coordinates & Stark effect

I am trying to solve the hydrogen atom in parabolic coordinates and find the first level correction of the Stark effect. The Hamiltonian is (free part + interaction): $$ H^0 = -\frac{1}{2}\nabla^2 - \...
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Probability of finding particle in ground state of old potential immediately after potential is changed?

Suppose a particle’s wavefunction satisfies the 1d time-independent Schrodinger equation with potential $U(x)$ and that its ground state is known to be $\psi_0(x)$. The particle is in the state $\...
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Why do we sometimes transfrom the interaction picture back into the Schrodinger picture?

We go first into the interaction picture, then say do a rotating wave approximation and then go into the rotating frame of the driving frequency, we solve the schrodinger equation and then and the end ...
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Can the initial wavefunction be discontinuous?

In a infinite potential well of width $a$, an electron starts in the left half and at $t=0$; it is equally likely to be found at any point in that region. To find the wavefunction at later times, we ...
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Coefficients of the wave function - a free particle in a box [closed]

If we solve the time independent Schrödinger equation for a particle in a box of length $L$, we get: $$\psi_n\left(x\right)=A\sin\left(\frac{\pi n}{L}x\right)$$ I then see that we normalize $A$ such ...
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Critique on various ways to think about time reversal transformation on Schrodinger equation?

Please define how time-reversal symmetry act on Schrodinger equation $i \frac{\partial}{\partial t} |\Psi{}(t) \rangle = H(t) |\Psi{}(t) \rangle.$ (for general form: which can be relativistic such as ...
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What is the difference between the time dependent and time independent Schrödinger equation?

I've already gone through a couple of questions regarding the Schrödinger equation and none of them seem to solve my doubt. Some say that the Time Independent Schrödinger Equation (TISE) is just a ...
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Is the linear combination of eigenfunctions of a time-independent Hamiltonian also a solution of the time independent Schrodinger equation?

Consider a system where the Hamiltonian is time independent, the wavefunction which is say a linear combination of the eigenfunction of the Hamiltonian (with different eigenvalues) is not the solution ...
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Number of states in the free electron gas

Considering the free electron gas model and the representation of stationary states in the k-space, the book I'm reading (Griffith's Introduction to Quantum Mechanics) says that "each ...
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Estimating ground state of Yukawa potential using a variational method [closed]

I have to calculate an upper bound for the ground state energy $E_0$ given the Yukawa potential $$ V(r) = -\dfrac{g}{r} e^{-kr}\ ,\quad g,k > 0\ , $$ and a test function family $$ \phi_\lambda (r) =...
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Time evolution of a density operator

$\newcommand{\ket}[1]{|#1\rangle}\newcommand{\bra}[1]{\langle#1|}$There is a known expression for evolution of density operator in time: $$\rho(t) = U(t,t_0)\rho(t_0)U^{\dagger}(t,t_0)$$ Let's denote $...
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Sakurai: Time evolution operator

How does Sakurai derive the infinitesimal time-evolution operator from scratch without Hamiltonian? $$\mathcal{U}(t_0+dt,t_0) = 1 - i\Omega dt.$$ It is definitely from Taylor's expansion. But complex $...
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1D infinite potential well

I'm studing quantum mechanics and I'm stuck on this problem. For 1D infinite potential well: $$V\left(x\right)=\left\{\begin{matrix} 0 \quad {\rm if} \quad 0<x<a \\+\infty \quad {\rm if} \quad x&...
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388 views

Adding a constant term to potential in Schrödinger's Equation

If we add a constant term $k$ to the potential function in time-independent Schrödinger's equation, $V(x) \rightarrow V(x)+k$, then how does it affect the solution, and what is its significance? ...
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Similarity in the solution for diffusion and Schrödinger equations [duplicate]

Both diffusion and Schrödinger equations are PDEs (first order in time and second order in space) with a different physical meaning. When solving a simple case of 1d diffusion with fixed boundary ...
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Why is the phase of a matter wave not Galilean invariant? And what does this say about the Schrödinger equation? [duplicate]

Matter waves are not Galilean Invariant Consider a non-relativistic freely-propagating matter wave in an inertial frame $\Sigma'$ moving along the $x'$-direction with kinetic energy $E'=1/2m_0v'^2$, ...

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