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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QM: Why is there a minus sign on the Energy operator when using complex conjugate?

I understand how they get the first equation. But I have no idea why there is a minus sign on the second equation: This is from a derivation for the probability density current found here: ...
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92 views

How to find the time evolution for two-component spinor?

Taking the wave function I found here as the initial condition for the Schrodinger equation. I would like to find how the wave function will depend on time, at least for small times, and calculate ...
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2answers
68 views

Why isn't the Time-Independent Schrödinger Equation an equation of motion?

I thought an equation of motion was something where you are given a Lagrangian and, using the Euler-Lagrange equation, you then find the equations of motion for that system. Same basic idea for the ...
2
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1answer
50 views

Help in solving Schrödinger equation for Hydrogen

I have almost finished getting the solution to the Schrödinger equation for the hydrogen atom (got the theta and phi component equations), but am stuck on the r component equation. Can anyone help me ...
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1answer
45 views

How do I take take the partial derivatives of the general solution to the TDSE for a free particle? [on hold]

Consider the general solution to the time-dependent Schrödinger equation for a free particle \begin{align*} \Psi(x,t) &=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{+\infty} \phi(k) e^{i\left(\hbar ...
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1answer
34 views

Can someone clarify what should and should not be an operator in my verification of the 1D solution to the SE for a free particle?

I just worked out the 1D free particle solution to the Schrödinger equation. My wave function was \begin{equation} \psi(x,t) = Ae^{i(px-Et)/\hbar} \end{equation} So I plugged this into both sides ...
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1answer
31 views

How do I find the average kinetic energy and average potential energy of a hydrogen electron in the ground state?

In my modern physics class, we are wrapping up the 3D Schrödinger equation, and I am more than a little lost. A few chapters ago, we learned about operators, and I have an equation for both these ...
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4answers
158 views

Derivation of Schrödinger equation - free particle

I learn quantum physics from Alonso-Finn's book (Amazon link), there's one step of Schrödinger equation for a free particle that I couldn't understand. $$ \frac{\mathrm{d^{2}\Psi } }{\mathrm{d} ...
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253 views

Numerical solution to Schrödinger equation - eigenvalues

This is my first question on here. I'm trying to numerically solve the Schrödinger equation for the Woods-Saxon Potential and find the energy eigenvalues and eigenfunctions but I am confused about how ...
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1answer
83 views
+50

Eigenvalues of the radial Schrödinger equation on a finite integration interval

There are numerous ways to estimate the eigenvalues of a radial Schrödinger equation, see http://arxiv.org/abs/math-ph/0703040 as an example. Anyhow, the formulas only cover the Schrödinger equations ...
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1answer
54 views

Energy levels in close-proximity of each other in time-independent degenerate perturbation theory

I've applied second order time-independent degenerate perturbation theory corrections to the energy with the method presented in Modern Quantum Mechanics by J.J. Sakurai. I shortly summaries this ...
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2answers
31 views

Interaction Hamiltonian in the interaction picutre

The Schrodinger and Heisenberg pictures make sense to me. But the interaction picture which is a hybrid of the two does not. Author of this text first splits the Hamiltonian up as ...
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3answers
152 views

Quantum Mechanincs - Dirac notation and solving time dependant schrodinger [closed]

The $\hat{S}_{x},\hat{S}_{y},\hat{S}_{z}$ obviously correlate to $x,y,z$ components of the operators. Consider the Hamiltonian: $$\hat{H}=C*(\vec{B} \cdot \vec{S})$$ where $C$ is a ...
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1answer
36 views

Centrifugal Term

I'm studying quantum mechanics with Griffiths (2 nd edition) and I have one question related to the Schrodinger equation in spherical coordinates. In the radial equation: ...
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38 views

WKB Quantization Condition - negative?

In deriving the quantization condition for a bound state in a potential with "no verticle walls" we start with the WKB connection formulas to find the wavefunction in the interior of the well ...
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1answer
40 views

How much time between measurements do you have in order to make the same measurement on a particle?

As I understand it, you can make a measurement on a particle and if you quickly carry out a second measurement you will get the same outcome as the prior measurement. If this is the case, how much ...
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2answers
89 views

How do we know $\psi$ depends on $n,l,m$

Regarding the separation of $\psi$ to an angular and radial part, why does each part have a specific dependence of the quantum numbers? How can Schrodinger equation describe a system just from its ...
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106 views

Triangular barrier in infinite potential well

Suppose I am looking to solve the wavefunction for the following 1D potential: $$U(x) = \begin{cases}V_0\frac{a-|x|}{a}&\quad\text{for}\quad|x|<a ...
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59 views

WKB formula and Langer correction [duplicate]

The general WKB approximation formula states that $$ \int_a^b\sqrt{E_k-V(x)} = (k+1)\pi \text{ with } x \in [a,b] $$ for a regular Schrödinger equation (without the $\hbar$ and such). However, in the ...
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47 views

Solving Schrodinger Equation with Anisotropic Effective Mass

How can I discretize a time-independent Schrodinger equation using the mass tensor and considering the valley degeneracy for the specific material at hand? I intend to investigate the confinement ...
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33 views

How to solve a difficult equation describing large vacuum fluctuations?

Suppose that a Quantum System can be described by the wavefunction $\psi(\vec{x},t)$, but due to the occurence of chaotic noise within the Quantum System, only the "filtered" wavefunction ...
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1answer
90 views

Can I state that $\Psi (x_1, \dots , x_n,t)= \sum_{i=1}^n a_i \psi (x_i,t) $ via superposition?

Given that the hamiltonian $\hat H$ of a system is a linear operator and $\dot \psi (x_i,t)$ does not depend on spatial coordinates $x_1, ..., x_n$ with bases $\hat e_1, ... , \hat e_n$ can I state ...
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1answer
33 views

Probability density function of a particle for computation [closed]

I'm writing a program, part of which relies on a particle being able to change location similar to a how a real particle would behave (pardon my physics). For example, on a grid of 100x100, a ...
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2answers
88 views

Is the formula for Schrodinger's equation on Wikipedia incorrect?

http://en.wikipedia.org/wiki/Schr%C3%B6dinger_equation#Time-dependent_equation On Wikipedia, the SWE contains a term called reduced mass. After consulting several peers, no one knows what this has to ...
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51 views

What is the purpose of knowing the value of ground state energy of a potential well?

Using the formula $$E ~=~ \frac{\pi^2\hbar^2}{2 m a^2}$$ where $a$ is the length of an infinite potential well. It is apparent that as $a$ get smaller i.e. from a metal to the size of an atom, the ...
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Solution to Schrödinger equation

I'm trying to solve the Schrödinger equation for a given potential. With some assumptions I end up with: $$\frac{\hbar^2}{2M}\frac{d^2u(r)}{dr^2} = - \left(E - V(r)\right)u(r)$$ Since it's a square ...
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48 views

What kind of potentials can be used in Schrödinger's equation?

I have a couple of questions about what kind of potentials can be used in Schrödinger's equation: How about the potential from a magnetic field? Isn't Dirac's equation more appropriate in that case, ...
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36 views

Asymptotic Analysis of 1-D Schrödinger Equation [closed]

I'm looking to do a small personal project regarding the time independent Schrödinger equation in 1-D: $$y'' +V(x)y=Ey$$ $$y''=Q(x)y$$ where $ Q(x):=E-V(x) $. There is obviously nothing stopping ...
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66 views

Can someone clarify which (if any) of these three QM assumptions is wrong?

I am trying to learn more about quantum mechanics. I am reading a book by Griffiths that I like. I'm trying to summarize what I've learned. So below I provided three assumptions. I'd like to know if ...
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43 views

Nodes of the ground state of a system of Schrödinger equations

In 1D, a single wave function that satisfies Schrödinger's equation representing the ground state for some $V(x)$ has no nodes. Suppose now that you have a system of $N\neq 1$ coupled Schrödinger ...
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208 views

Normalizing the solution to free particle Schrödinger equation

I have the one dimensional free particle Schrödinger equation $$i\hbar \frac{\partial}{\partial t} \Psi (x,t) = -\frac{\hbar^2}{2m} \frac{\partial^2}{\partial x^2} \Psi (x,t), \tag{1}$$ with ...
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23 views

Solving the quantum well gives you eigen energies gives $E_n$, are these energies in conduction band or valence band?

I wonder if the energies $E_n$ that is derived from solving the SWE for the quantum well can be considered as energies in the conduction band or the valence band. In other words is $E_1$ is lowest ...
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1answer
50 views

Losing a term for 3D radial schrodinger equation

I am trying to solve the Schrodinger equation For a potential $V(r)$ defined for $ 0<r<R$ as $$V(r)=-V_0 $$ and zero everywhere else. For wavefunction $u$ I can easily get to $$ u'' =-k^2u,$$ ...
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39 views

Is it ever appropriate to write $| \phi(t)>$

I am trying to solve the Schrodinger's wave equation $\hat H |\psi(x,t)> = E|\psi(x,t)>$ using separation variables so that $\psi(x,t) = \psi(x)\phi(t)$ Solving the equation involves the step ...
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1answer
105 views

Checking that the propagator for Harmonic Oscillator satisfies Schroedinger Equation [closed]

I have the propagator for the harmonic oscillator. $$K(x_f,x_0,t)=\sqrt{\frac{m\omega}{2 \pi \hbar \sin{wt}}}\exp\left(\frac{i}{\hbar}\frac{m\omega}{2 \sin{\omega t}}((x_0^2+x_f^2)\cos\omega ...
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1answer
82 views

Schrodinger equation, commutative operators, and Symmetry

When solving Schrodinger's equation in 3D with a spherical laplacian you reach a point at which you introduce a separation constant and can see that the same eigenvalue satisfies the radial and ...
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1answer
127 views

Is the expression $S=K \log(\Psi)$ appearing in Schrödinger's first paper well defined?

I am currently reading Schrödinger's papers and happen to have some questions that maybe some expert in the field could clarify for me. Like what happens with $$S = K \log(\Psi)$$ when $\Psi<0$. ...
6
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3answers
556 views

Why is the Klein Gordon equation of second order in time?

I was wondering if there is any way to interpret the fact that the Klein Gordon equation is a 2nd order PDE in time. I mean, normally you would expect that as soon as you fix the initial wavefunction, ...
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31 views

Exercise about Bethe Ansatz for $N=3$ particles on a ring of length $L$

Suppose there are $3$ bosons living on a 1-dimensional ring of length $L$. The Hamiltonian is given by $$H=-\sum_{i=1}^3\frac{\partial^2}{\partial x_i^2}+\sum_{1\leq j<k\leq ...
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88 views

Is hydrogen atom in a box solvable analytically?

Schrödinger's equation for hydrogen atom in free space can be easily solved by switching to center of mass frame, introducing reduced mass and separating variables in the resulting 3D problem. But ...
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39 views

How to compute minimum shallowness of quantum well to have at least one bound state?

Given a potential $V$, how does one compute how shallow the potential can be such that it allows at least one bound state?
2
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1answer
49 views

Is there a mathematical explanation for why there occur bound states if the effective potential falls below zero?

Usually in physics textbooks, if the effective potential of the radial schroedinger equation $$-\frac{d^2}{dr^2}u(r) + \frac{\ell(\ell+1)}{r^2}u(r) + V(r)u(r) = E u(r)$$ falls below zero in some ...
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51 views

Basis spin states

We are given a system of $N$ spin states and the following (non-hermitian) Hamiltonian $$H = \frac{N \hbar \nu}{2M} \sin(\alpha)+ \sum_{i=1}^N \frac{\hbar \omega_i }{2} \sigma_{z,i} + \frac{\hbar \nu ...
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124 views

Physical meaning of linear combination of possible states in infinite well

The solution of infinite well, positioned from $x=0$ $x=l$, is $$ \Psi_n(x,t)= \sqrt{\frac{2}{l}}\sin\left(\frac{n\pi}{l}x\right)e^{iE_nt} $$ But the most general solution of this problem is : $$ ...
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Renormalization of perturbation theory in non-relativistic quantum mechanics

A simple example: In calculating the nonlinear polarization of an atom, in perturbation theory, we typically get something like: $$p^{(n)} = \sum_{m=0}^n \left< \psi^{(m)} \right| \mu \left| ...
3
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1answer
80 views

Estimate for minimum potential depth required for bound states in 3D attractive potential well

Consider a 3D spherically symmetric potential well, $$H = \frac{p^2}{2m} + V(r)$$ with $V(r) = - V_0$ for $r < a/2$ and $0$ else, for some $V_0 > 0$. Now, it is well known that $V_0$ needs to ...
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1answer
90 views

Must the wavefunction be analytic?

In order to show the preservation of normalization of the wave function (in one dimension for now), one shows that the time differential is zero, which entails the following step: $$ ...
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1answer
69 views

Realistic Potential Wells

What is meant by the term "realistic" potential wells? I got stuck into the term as I don't know what are the limitations of the word realistic in this case. For example mentioned in line We ...
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87 views

Why don't the De Broglie dispersion relation contain a constant term?

Wikipedia says that the dispersion relation for a non-relativistic particle is: $$ \omega = \frac{\hbar k^2}{2m}. $$ But when I tried to calculate it myself, I seem to get a constant term in that ...
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Does “Schrödinger equation” have practical usage? [closed]

I'm not physicist and sorry if a question is a bit strange, but for me when I look to Schrödinger equation it seems that it is more philosophical than mathematical. For example, Newton's law of ...