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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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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33 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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33 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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78 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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83 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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53 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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40 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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40 views

Perturbation theory of states [on hold]

A particle in 1 dimension with mass $m$ is in potential well with $V = 0$ for $–a/2 < x < a/2$ and infinite potential elsewhere. The particle is initially in the state $n = 10^9 + 1$, with the ...
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30 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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85 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
24 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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81 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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49 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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74 views

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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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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33 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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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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39 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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156 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
48 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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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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92 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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76 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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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 $\Psi<0$. ...
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535 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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28 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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84 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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34 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?
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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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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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4answers
109 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| ...
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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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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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68 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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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 ...
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Quantum Mechanical Thinking

I've just been wondering about how atoms and molecules can be quantum mechanically thought about, and I have a question. It is often said that intermolecular bonding is purely "electrostatic". I hope ...
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179 views

Given a wave function $\psi(x)$, is there always a potential $V(x)$ such that $\psi(x)$ is an eigenstate?

Given any unit norm wave function $\psi(x)$ which is in the Hilbert space, can we always find a $V(x)$ such that the $\psi(x)e^{-i\omega t}$ is a solution of the corresponding Schrödinger equation? (I ...
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272 views

Who is doing the normalization of wave function in the time evolution of wave function?

In the Schrodinger equation, at any given time $t$ we should jointly add another sub equation, like $$||\psi_t(x)|| = 1$$ where $\psi_t(x) = \Psi(x,t)$, and then try to solve the two equations ...
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Physical examples of wave function that is stationary in space and varying in time. Something like $\Psi(x,t) = \psi(t)e^{-ikx}$

We know time independent Schrodinger equation, where the wave function is stationary in time and varies in space. Simple examples are, particle in an infinite potential well, and the Hydrogen atom, ...
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Is it possible extend Schrodinger theory in relativistic contexts with naive consideration?

Preamble Let's consider a generic sinusoidal wave $\Psi (\mathbf{r},t) = A e^{i(\mathbf{k} \cdot \mathbf{r} - \omega t + \phi)}$ and let's insert it into Schroedinger equation (please note that $ ...
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Quantum Mechanical Wave Functions

Are wave functions, such as those used in the Schroedinger equation just 'guessed' and verified, or are there other theories which tell us the mathematical description of the wave function for ...
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General question about the potential barrier problem: Why does $\exp( kx)$ diverge when $x>0$ in the case when $E < V(x)$?

For the two images below, the first potential barrier has particles approaching it where $E > V_o$ & the second has a particle that has $E < V_o$, where $E$ is the energy of the particles ...
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60 views

Boundary Conditions in a Step Potential

I'm trying to solve problem 2.35 in Griffith's Introduction to Quantum Mechanics (2nd edition), but it left me rather confused, so I hope you can help me to understand this a little bit better. The ...
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What is the implication that the Schrodinger equation be solved by both real and imaginary part of the wave function? [closed]

Suppose $\psi = \psi_{real} + i \psi_{imag}$ be the wave function, then both $\psi_{real}$ and $\psi_{imag}$ can be used to solve the Schrodinger's equation This can be demonstrated by plugging ...
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1answer
73 views

What is the energy of a Gaussian wave packet?

Suppose we have a potential barrier situation, that is $V(x)$ is zero everywhere except on the interval $[-a,a]$, where it is equal to some $V_0 > 0$. Introduce some Gaussian shaped wave packet to ...
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1answer
109 views

What are the relative limitations of the Schrödinger, Pauli, and Dirac Equations?

I know there are significant differences in the nature of the Schrödinger, Pauli, and Dirac equations. Although I know a bit about how each works, I don't understand the relative limitations of each ...
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130 views

Schroedinger Equation and Special Relativity

From what I understand, the Schroedinger equation describes how the wave function of a quantum system evolves in space over a given time (I am referring to a relativistic version of the Schroedinger ...