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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Derivation of the Klein-Gordon equation from Schrodinger equation [on hold]

Can someone demonstrate how to transform the one dimensional Schrodinger equation, $$-\frac{\hbar^2}{2m}\frac{\partial^2}{\partial x^2}\phi = i\hbar\frac{\partial}{\partial t}\phi$$ into ...
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75 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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24 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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41 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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46 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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4answers
92 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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19 views

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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47 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
81 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
62 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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1answer
69 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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71 views

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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64 views

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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2answers
161 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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218 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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1answer
80 views

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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144 views

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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1answer
60 views

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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77 views

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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1answer
39 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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52 views

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
56 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
97 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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114 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 ...
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50 views

Cubic perturbation to coupled quantum harmonic oscillators

I recently came across this two-dimensional problem of a particle in a potential of the form $$V = \displaystyle{\frac{1}{2}m \omega^2} \big(y^2 + x^2y \big) - \alpha y,$$ where $x$ and $y$ are known ...
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1answer
54 views

using a given wavefunction to find particle properties

Let's say we have a given wavefunction and we want to find a particle that will fulfill the properties for that wavefunction. How can we do that? Is it possible? I was thinking of using Schrodinger's ...
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3answers
589 views

How to motivate Schrödinger's Equation? [duplicate]

Schrödinger's equation is supposed to be a differential equation for the wave function of a particle. As I currently understand, De Broglie's hypothesis is a hypothesis that for particles there should ...
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22 views

Applications of the study of Hamiltonians with constant magnetic fields

I am interested in understanding possible applications for the study of quantum systems with constant magnetic fields. For definiteness, consider the Landau Hamiltonian $$H_{0} = ...
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1answer
32 views

Allowed energies for semi-harmonic oscillator

Question: If a particle is attached to a semi-harmonic oscillator (that is, for example, the spring is stretchable but not compressible) such that the potential $V(x)$ is infinity for $x\leq0$ and ...
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3answers
64 views

Schrödinger: Coherent states

A coherent state is called $\Psi_{{\alpha}} \left( x,t=0 \right)$ and is defined by: $a_{{{\it \_}}}\Psi_{{\alpha}} \left( x \right) =\alpha\,\Psi_{{\alpha}} \left( x \right) $ where $a_{{{\it ...
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Wave-Particle Duality in the Confinement of an Electron in a Box [closed]

According to the wave particle duality, one can say that an electron is both a wave and a particle. If we confine it in a box, it can only form standing waves at particular wavelengths, which leads ...
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1answer
89 views

Complex comjugate of Schrodinger equation: paradox in matrix form?

We can take the complex conjugate of schrodinger equation, and obtain $$ -\frac{\hbar^2 }{2m}\frac{\partial^2\psi}{\partial x^2} + V(x)\psi = i \hbar \frac{\partial \psi}{\partial t} $$ $$ ...
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2answers
58 views

How are electrons restricted to individual orbitals?

Since orbitals are just regions of electron density, they allow electrons to occupy the same space. I feel like in some sense this contradicts the Pauli exclusion principle limiting two fermions, or ...
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Low-momentum behaviour of a short-range potential

I've read the follow sentence in a journal article (arXiv preprint) which constitutes part of my coursework. As discussed in the previous section, the low-momentum behavior of any short-range ...
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22 views

What's the elementary reason for the formation of a band gap?

Bohr's solution for an isolated hydrogen atom showed that there are only discrete allowed energy levels, $E_n = -{E_0\over n^2}$ and the solution of the Schreodinger equation provided a certain ...
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1answer
56 views

Naive interpretation of Galilean invariance of the TDSE

I was told today by someone smarter than myself that the time-dependent Schroedinger equation in one dimension was invariant under a Galilean transformation of $(x,t)$, namely under ...
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1answer
129 views

Quantum Spin Simulation

In Leonard Susskind's Quantum Mechanics: The Theoretical Minimum, he describes a computer program that could fool you into thinking there is a quantum spin in a magnetic field. This spin is inside a ...
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86 views

Physical interpretation of a certain Hamiltonian

Consider a $2 \times 2$ Hermitian (or symmetric) matrix-valued function $$g(x) = \{ g_{jk}(x)\}_{j,k=1,2}, \quad x \in \mathbb{R}^{2},$$ such that $0 < m_{-}I \leq g(x) \leq m_{+}I$, for some ...
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Estimate of the second shallowest bound state?

Suppose we have a 1D potential $V(x)$ of finite range, i.e., $$ V(x) ~=~0 $$ for $|x| > b $. The potential is assume to support at least two bound states, but might have more, say $n\geq 2$. ...
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2answers
313 views

How to know if a wave function is physically acceptable solution of a Schrödinger equation?

How does one decide whether a wave function is a physically acceptable solution of the Schrödinger equation? For example: $\tan x$ , $\sin x$, $1/x$, and so on.
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152 views

1D Finite potential well: solutions with $\sinh$ and $\cosh$?

So I am studying the (one dimensional) quantum mechanical finite potential well defined by: $$ V(x) = \cases{0, &|x|>a\cr -V_0, &|x|<a} $$ where $V_0>0$ is a real number. I know ...
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2answers
92 views

Collapse of the wave function and Heisenberg uncertainty

I have been studying quantum mechanics for a few weeks, in particular wave mechanics, as created by Schrodinger, and his equation. As a high school student, I haven't found an answer to this question ...
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1answer
117 views

Deriving a Useful Solution of the Schrödinger Equation [closed]

How does one derive the fact that $$\psi(t,x) = (\tfrac{2 \pi \hbar t}{m})^{-d/2}\int_{\mathbb{R}^d} e^{im\tfrac{(x-y)^2}{2\hbar t}}\psi_0(y)dy$$ is a solution of the time-dependent Schrödinger ...
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Why discrepancies in the Schrödinger equation? [duplicate]

Why is there seemingly two definitions of the Schrödinger equation? \begin{equation} i\hbar\frac{\partial}{\partial t}\Psi=\hat H\Psi. \end{equation} And \begin{equation} i\hbar ...
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Quasiclassical QM for central fields

Let's have quasiclassical QM for central field $V(r)$. The Schroedinger equation for radial part of wavefunction $R_{nl}$ after substitution $u_{nl} = rR_{nl}$ takes the form $$ u_{nl}{''} + ...
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86 views

What is the most agreed upon quantum mechanical equation of motion?

On multiple Wikipedia articles, it mentions several quantum mechanical equations of motion, namely those by Schrödinger and Heisenberg. Which one is the most accurate and agreed upon quantum ...
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120 views

A misunderstanding regarding infinite square well

Here is a picture of the energy states of infinite potential well. We can see That the first level have a half wavelength which fittes with a full wave of the second level. $$\frac{ \lambda _{1} ...
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161 views

Ground state of Spherical symmetric potential always have $\ell=0$?

I was given a problem where I have a spherically symmetric potential (the exact form is not relevant to this question, I think - but anyway is it 0 for $r\in[a,b]$ and $\infty$ everywhere else) and I ...
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2answers
63 views

Semiclassical quantization of bouncing ball

Consider an elastically bouncing ball of mass $m$ and energy $E$. This has a triangular potential $$ V(x)~=~\left\{\begin{array}{ll} mgx & \text{if } x>0, \\ \infty & \text{if } x<0, ...
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25 views

Quantization in a 2D quasicrystal?

let be the Schroedinger equation with a potential in a quascristal $$ -( \partial _{x}^{2}+ \partial _{y}^{2}) \Psi (x,y)+ V(x,y)=E_{n} \Psi (x,y) $$ if we were in a crystal we could impose the ...