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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Lagrangian for Schrödinger field equation [duplicate]

I know this topic has already appeared in some posts but I think my question is uncovered. Please correct me if I'm wrong. I'm trying to find a lagrangian for the Schrödinger field equation $$i \frac{\...
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Infinite linear potential well?

I am currently learning and solving Schrodinger's time independent equation for particles under various 1D-potentials. Would it be possible to have a mix of a linear potential (of the form $U(x)=Fx$ ...
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Why is the first derivative of the time-dependent Schrödinger equation continuous? Where does it come from?

I was taught in first year physics that the first derivative of the time-dependent Schrödinger equation had to be continuous. However I was never taught (or at least I don't remember) the reason why. \...
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What is the physical justification for the boundary conditions of the Schrödinger equation for an infinite potential well?

All the literature says that the physically meaningful solutions to the Schrödinger equation in an infinite potential well must fulfill the boundary condition that the wave function is $0$ at the ...
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How does the linearity of the Schroedinger equation reflect the interactions?

There is a common lore that linear equations describe non-interacting systems, why non-linearities correspond to non-trivial interactions. My (loose) question is how is that compatible with the ...
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Why does the wavefunction of a particle spread out after a measurement?

Quantum mechanics states that the wave packet of a particle "spreads-out" in position again after a measurement on this particle has been made. Is this spreading or "dispersion" ...
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Proof that infinite square well gets split into two by a Dirac delta function in the middle

I was reading these lecture notes https://faculty.biu.ac.il/~barkaie/TA6.pdf , and at the end, they claim that $\tan{kL}=0$ implying $kL=n\pi$, thus resulting in the same condition as before, means ...
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What does it mean that a free particle has no definite energy in quantum mechanics?

In the quantum mechanics case of the infinite square well, the general solution to the Schrodinger equation is a linear combination of solutions with definite energy states. When you measure the ...
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How is the Ehrenfest theorem satisfied in a stationary state?

The Ehrenfest theorem in quantum mechanics for a particle moving in one-dimension in an arbitrary nonuniform potential $V(x)$ is $$\frac{d}{dt}\langle p\rangle=-\left\langle\frac{\partial V(x)}{\...
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What determines the Hamiltonian of an isolated quantum system?

Let an isolated quantum system be in state $|\psi\rangle$. Then, quantum mechanics says the system evolves in time according to some Hamiltonian $H$, which does not depend on $|\psi\rangle$. But the ...
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What are the necessary and sufficient conditions for a wavefunction to be physically possible?

Often times it is stated in books that a quantum state is physically realizable only if it is square integrable. For example in Griffiths (2018 edition) page 14 he stated Physically realizable states ...
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Hilbert space in Dirac's representation

I've been reading some old posts here on physics stack exchange and I realized something that have never ocurred to me before. Let $\mathscr{H}$ be a Hilbert space over $\mathbb{C}$. An orthonormal ...
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Time dependent wave function of a particle in a gravitational field

I found this great question about the solution of the Schrodinger equation for a particle in a constant gravitational field, but the solution they wanted is to the time independent Schrodinger ...
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$V(x)=V_0\sin^2(k_0x) $ potential, 1 dimensional [closed]

If I have a potential like this: $$V(x)=V_0\sin^2(k_0x) $$ where $k_0^2=2m/h^2E_*$ how can I prove that the wavefunctions are: $\Psi_q(x)=e^{iqx}u_q(x)$ where $qe[-k_0,k_0]$ and $u_q(x)$ is periodic ...
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Set of probabilities distributions inside a infinite square well [duplicate]

Its been some years since I did the infinite square well. I am doing an econimics problems with probability distributions and I vaguly remember there being a name for either the wave functions in the ...
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How to derive the ansatz for even solutions of delta peak in infinite square well? [closed]

Consider the following potential: The resulting Hamiltonian will commute with the parity operator ($[\hat{P}, \hat{H}] = 0$) and thus solutions must be either even or odd. Now the delta potential ...
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Why use a path integral if we have a general solution to the Schrödinger equation? [closed]

In this answer, the general solution to the Schrödinger equation is given, and is also included here. In my QM class we talk a lot about this equation, but we haven't seen path integrals yet, and I ...
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Proof bound states exist on vanishing, negative potentials in 1D

I want to prove the following: Show that for any one dimensional, time independent potential $U(x)$, where $\lim\limits_{x \rightarrow \infty}{U(x)} = 0$ and $\int_{-\infty}^{\infty}U(x) < 0$ ...
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Eigen-energy in Finite Quantum Well

So I'm a beginner at quantum mechanics and I'm learning about finite quantum wells. I've been stuck on an example on how to find Eigen-energies in conduction and valence bands of the quantum well ...
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Energy Levels in Double Finite Well potentials [closed]

Why is there a behaviour where $E_1$ and $E_2$ are close to each other? This question is already answered before but I did not understand the explanation about symmetric and asymmetric terms as I do ...
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Slater determinant as (zero order) approximation to Schrodinger equation

I understand how the Slater determinant for N particle works and its application in writing an antisymmetric wavefunction. However the question asks me to show that the Slater determinant is a zero ...
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Solving the matrix Schrödinger equation

One can solve the Schrödinger equation by diagonalizing the Hamiltonian $H$. Due to limited memory, we truncate $H$ up to $N$. Now I red here on slide 8, that increasing $N$ cant lead to higher ...
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Showing that two wavefunctions are orthogonal [closed]

I have two function: $\Psi_A(x,0)=\sqrt{\frac{1}{6}}\phi_0(x)+\sqrt{\frac{1}{3}}\phi_1(x)+\sqrt{\frac{1}{2}}\phi_2(x)$ $\Psi_C(x,0)=\sqrt{\frac{3}{8}}\phi_0(x)+\sqrt{\frac{5}{8}}\phi_2(x)$ I have to ...
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When is the Schrodinger equation applied?

Does the Schrodinger equation require a trapped electron, electron bound to potential, to work? In such a case, how do we treat free electrons? Do we just use simple classical formulae?
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How are linear combination of energy eigenstates solutions to the TISE?

My textbook states that: TISE is a generic eigenvalue problem and the superposition state is not a solution to the TISE. Another part of textbook (in context of infinite square well): The most ...
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Finite Quantum Well [closed]

I want to start by saying i promise i'm not stupid but this has me stumped for more than a week now. I have to find the number of Quantum states in the conduction band. Easy enough $V_0 = \Delta{Ec}= ...
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How to solve double delta potential bound states by “brute force”

I just solved a problem in Griffiths' Intro to QM, where one had to find the bound states given the potential: $$V(x)=-\alpha [\delta (x-a)+\delta(x+a)]$$ In order to solve it, one had to exploit the ...
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Why energy restrictions in the infinite square well are general?

I'm looking at the infinite square well case for solving the Schrodinger equation in quantum mechanics. I see that when solving the time-independent Schrodinger equation, we find that the energies: $$...
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Are atoms' most precisely known electronic transition frequencies determined theoretically or experimentally?

In principle, the electronic transition energies/frequencies for a given species of atom can be calculated by solving the time-independent many-body fermionic Schrodinger equation for $n$ electrons in ...
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Time-dependent Schrieffer-Wolff Transformation

The time-independent Schrieffer-Wolff Transformation has the form: $H' = e^{S}He^{-S}$. I see that the time-dependent form is: $H' = e^{S}He^{-S} +i\hbar\frac{\delta}{\delta t}(e^{S})e^{-S}$. Why ...
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Probability to find a particle inside a finite potential well

Supoose we have a repuslive potential well ($+V_0$) and, if a particle of energy $E(>V_0)$ is incident toward it, then it will feel repulsion, so intuitively the probability of finding particle ...
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Mysterious factor 2 in the Schrödinger equation derived from Feynman's Kernel

Feynman's Quantum Mechanics and Path Integral has a vivid physical interpretation of the path integral formalism. But I was stumbled on some mathematical details while following his derivation of the ...
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Potential well with 2 delta potentials

I have a problem to derive transcendental equation for eigenenergies of bounded states for potential: $$ V(x)= \begin{cases} -\lambda\delta(x+a/4)-\lambda\delta(a-a/4), & |x|<a/2 \\ ...
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Time-independent constant-Hamiltonian Schrodinger equation solution when at time 0 there's a matrix with mostly 0s?

So, I'm trying to model the behavior of a particle in a 2D optical lattice. I've done it successfully for 1D. I'm using the time-dependent Schrodinger equation with a constant Hamiltonian, so $\Psi(t)=...
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Probability density for free gaussian wave packet

I am doing an exercise where the wave function for t>0 is: $\psi (x,t) = \frac{e^{\frac{-x^2}{4a^2(1+\frac{it}{\beta })}}}{(2\pi)^{\frac{1}{4}}\sqrt{a} \sqrt{1+\frac{it}{\beta }}} $ with $\beta =...
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Bound state of Hydrogen atom at large $r$

When the radial equation of SE is solved for Hydrogen atom, to see the asymptotic behavior, we assume $r$ tends to infinity. The differential equation we are left with is: $$ d^2U/dr^2 = -\frac{2mE}{\...
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How were the frequencies of the Zeeman effect derived?

The Zeeman effect in quantum physics is basically when electrons feel different effects of an external magnetic field (due to their different orientations of angular momentum), thus electrons making a ...
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What does having an imaginary part of potential imply? [duplicate]

In DJ Griffiths' ''Introduction to Quantum Mechanics" to describe an unstable particle that spontaneously disintegrates he assumed an imaginary part in the potential. What does that signify? What ...
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Is the energy of the potential step quantized?

I'm solving the Schrödinger equation for a potential step and I was wondering if the energy of the potential step is quantized?
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Eigenvalue and Eigenfunction for a particle trapped in a 1D infinite asymmetric potential well [closed]

As we're barely scratching the surface of Quantum Physics in class, we haven't been taught about asymmetric potential wells. However, I find it fascinating, moreover difficult, to find the eigenvalues ...
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Comparing the quantum version of the infinite well to classical version

In the quantum version of the infinite well, the energy eigenvalues can be precisely determined. The energy is all in the form of kinetic energy, $E=\frac{p^2}{2m}$, and so, classically, the magnitude ...
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Principal quantum number of the classical particle

The example 7.9 in this page shows the principal quantum number of the classical particle. A small 0.40-kg cart is moving back and forth along an air track between two bumpers located 2.0 m apart. We ...
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Position of a particle in the standard infinite well [closed]

If one made many measurements of the position of a particle in the standard infinite well and binned the result to estimate $P_{CL}(x)$, what would the result look like? What would it look like for a ...
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Classical & weak solutions of Schrödinger equation [closed]

Consider the problem of an infinite square well $$ V(x) = \begin{equation} \begin{cases} 0, \qquad {\rm if}\quad0 \le x \le L \\ \infty, \qquad{\rm otherwise} \end{cases} \end{...
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What motivates the trial solution of $\left[-\frac{\hbar^2\nabla^2}{2m}+\frac{e^2y^2B^2}{2m}-\frac{i\hbar eyB}{m}\right]\psi(x,y,z)=E\psi(x,y,z).$?

The time-independent Schrodinger equation for the problem of charged particles in an uniform magnetic field ${\vec B}=B{\hat k}$, in the Coulomb gauge ${\vec A}=(-yB,0,0)$, reduces to the following ...
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Eigenfunctions and eigenvalues of particle in 2D box

A particle in a 2D potential box has two degrees of freedom. It is bound by the infinite potentials at the boundaries. Our professor asked us to resolve this into its respective $x$ and $y$ components,...
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How do I find the wave function in a separable Hilbert Space?

I am confused as to how I would go about finding the wave function in the Hilbert Space. As I understand, a wavefunction in the Hilbert space can be represented as $$|\Psi\rangle = \sum_{n} c_n|\psi_n\...
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The expectation value of momentum in an infinite well stationary state [duplicate]

For a particle in an infinite well potential given by: I am able to successful derive the normalized wavefunction as: $$u_n(x) =\sqrt{2/a}\sin(\frac{n\pi x}{a})$$ where this normalization has the ...
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Spin $\frac{1}{2}$ particle in a box with discontinuous magnetic field

Consider a spin $\frac{1}{2}$ particle bound in a one-dimensional box $-a<x<a$: its eigenfunctions should then be the common eigenfunctions of the free particle in the box, times the ...
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Why wave function does not vanish at a Dirac delta potential?

I have studied that a wave function should vanish at the location of an infinite potential. Consider a direct Delta delta potential at $x=0$. Why does does function not become zero here at $x=0$?

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