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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Do the symmetrized solutions to the time-independent Schrodinger equation span the symmetric functions?

Im learning QM from a book and after reading about multiparticle systems (for instance two noninteracting particles), it looks like the author found the solutions to the time-independent Schrodinger ...
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Why is an eigenfunction $\psi_{n,l}$ proportional to $r^{l}$ close to the nucleus?

This is in reference to Quantum Physics of Atoms, Molecules, Solids, Nuclei and Particles by Robert Eisberg and Robert Resnick. The author writes Inspection of the eigenfunctions listed in table 7 - ...
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What is the probability of measurement in QM, dependent on time?

Consider a QM system with an observable $A$ and orthonormal eigenbasis $\{|n\rangle,n=0,1,2,\ldots\}$. Then we know that if the system is in some state $|\psi\rangle$ and we measure $A$, the ...
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Minimum energy eigenvalue [duplicate]

Why is the energy eigenvalue is always greater than minimum potential for a particle moving in a certain potential?
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Fundamental state and its energy given a potential $V(x)$ [closed]

I have to answer same questions about this potential: $$V(x)= \begin{cases} 0 & x\in[0,2a]\\-\lambda\delta(x-a) \\+\infty & otherwise \end{cases}$$ Are there proper eigenfunctions and proper ...
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Asymmetric potential well and discrete energy levels

Let us consider an asymmetric potential, which is piecewise defined as $V_1$ for $x<0$, $0$ when $0<x<a$ and $V_2$ for $x>a$, together with the condition $V_1 > V_2 >0$. In the first ...
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On solutions of Schrödinger equation

Consider a quantum system described by the wave function $\psi({\bf x}, t)$ and subjected to a time-independent ordinary potential $V({\bf x})$. The relative Schrödinger equation takes the form: $$\...
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Odd function (probability density function) describing the measurable quantity of a quantum particle [closed]

This question is in reference to Quantum Physics of Atoms, Molecue, Solids, Nuclei and particles by Robert Eisberg and Robery Resnick. The setup for an infinite square quantum well with the potential ...
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81 views

Superposition of potentials

I'm trying to understand what superposition of potentials means. For example, let be $$V_0(x) = \begin{cases} 0 &x \in [0,2a]\\ +\infty & \text{otherwise}\end{cases}$$ and $$V_1(x)=-\lambda\...
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Why is it selectively OK to rely on intuitive analogues when solving problems in Quantum Mechanics, such as the step potential problem?

I was looking at solved example (3.13) in the Schaum's Series book on QM by Yoav Peleg et al (2nd edn), where they solve for a step potential where a high energy particle is coming from the left and ...
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Multiple-parameter Schrödinger equation, such $U(\lambda_1,\lambda_2,\lambda_3)=e^{ -i \lambda \cdot \sigma }$

One can derive the Schrodinger equation as follows: $U(\delta t)|\psi(t)\rangle = (I - i \delta tH)|\psi(t)\rangle = |\psi(t+\delta t)\rangle \rightarrow i\frac{|\psi(t+\delta t) - |\psi(t)\rangle}{\...
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What interval to use when proving orthogonality of wavefunctions?

When proving that $\psi_1=\sin(n\pi x/a)$ and $\psi_2=\cos(n\pi x/a)$ are orthogonal to each other in a 1D box, the main problem that I am facing is what to use as the domain of integration. If I take ...
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What is the difference between the Interaction picture (Dirac Picture) and a rotating reference frame?

In David McIntyre's Quantum Mechanics, we examine an electron within a magnetic field $$\vec{B}=B_o \hat{z}+B_1[\cos(\omega t)\hat{x}+\sin(\omega t)\hat{y}]$$ The Hamiltonian is then time-dependent ...
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Where's the rest of it? - Schrödinger equation for a free particle whose position is known

Assumptions I will be working in one spatial dimension throughout. It should be possible to generalise to three dimension with relative ease but doing so adds little to the conversation, only serving ...
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Quantum Mechanic: Sperically symmetric finite well [duplicate]

Consider the following 3D potential: $$V(r)=\begin{cases}-V_0 \ \ \ \ \ \ & r \leq a\\ 0 &r > a\end{cases}$$ we want to find the eigenfunctions for $\ell=0$, in particular we are interested ...
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A simple doubt regarding the propagator of the free particle

I start by saying that I'm not a physicist so forgive me if this question is somehow trivial. I am studying the White noise approach to Feynman integrals, from Hida et al.'s "White noise analysis&...
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$\hbar$ in Schrödinger's equation

We all know Schrödinger's equation $$\mathrm{i}\hbar \partial_t |\Psi\rangle = H|\Psi\rangle$$ I'm trying to figure out why we multiply by $\hbar$ instead of e.g. $h$. What is causing us to ...
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Question about Math of Superposition

all. I am currently reading through a quantum mechanics book, and I was struggling to understand an equation presented in the review of the mathematics. The part where I am discusses one-dimensional ...
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Understanding the solution of the infinite spherical well

I have been reading Griffith's Introduction to Quantum Mechanics, and I just went over the solution of the infinite spherical well. He gives it as $$\psi _{nlm}(r,\theta, \phi) = A_{nl}j_l\left(\beta_{...
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Constancy of Wronskian when potential has Finite Discontinuities

I am working on a problem where for the 1-dimensional Hamiltonian $\hat{H}=(-1/2m)(d^2/dx^2)+\mathcal{W}(x)$ with $\mathcal{W}$ assumed to be smooth and real and $\Psi$ as solution to $\hat{H}\Psi=E\...
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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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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 [duplicate]

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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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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$k$-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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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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57 views

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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