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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Decay of excited electron from Schrödinger equation?

I'm currently studying for a quantum mechanics exam. I found some old exams which has the following question I'm trying to solve. Please tell me if my reasoning is wrong. Can you see from the ...
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2answers
124 views

Infinite square well that suddenly decreases in size

A well known exercise in basic quantum mechanics is the sudden (diabatic) increase of the length of an infinite square well. Now consider a particle in an eigenstate of an infinite well that is ...
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1answer
45 views

Barycenter and relative coordinates for schroedinger equation of the hydrogen atom

Heyho, i just realized i am not sure how one gets from: $\Big(-\frac{\hbar^2}{2m_e} \Delta_{r_e} - \frac{\hbar^2}{2M_P} \Delta_{r_p} +V(r) \Big)\Psi(r_e,r_p) = E \Psi(r_e,r_p)$ to: ...
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3answers
484 views

Schroedinger equation for hydrogen atom

I have got a problem understanding the meaning of the Laplace operator in the Schrödinger equation for the hydrogen atom. $$\Big(-\frac{\hbar^2}{2m_e} \Delta_{r_e} - \frac{\hbar^2}{2M_P} \Delta_{r_p} ...
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3answers
359 views

Bound states of the $V(x)=\pm \delta'^{(n)}(x)$ potential?

The $\delta(x)$ Dirac delta is not the only "point-supported" potential that we can integrate; in principle all their derivatives $\delta', \delta'', ...$ exist also, do they? If yes, can we look for ...
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24 views

History of delta barrier in QM [migrated]

I'm interested in finding something out about the history of the problem of the delta potential barrier in QM. Which was the first study to propose this problem, and perhaps any particular motivation ...
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46 views

The propagator method versus Schrodinger time dependent equation [closed]

What advantage does the the propagator, $K(x,t; x',t')$, method has over the solution of Schrodinger time dependent equation (STDE) for example in harmonic ocillator problems? I don't see any. See ...
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1answer
80 views

Why isn't the time-derivative considered an operator in quantum mechanics? [duplicate]

Based on my understanding when doing quantum mechanics we deal with a small set of mathematical objects: namely scalars, kets, bras, and operators. But then in the Schrodinger equation we have this ...
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47 views

Evolution of a 'state' in the Heisenberg picture

Suppose that we have a Hamiltonian, $\hat{H}$, and an operator $\hat{A}$ which satisfies the Heisenberg equation$^{[a]}$ $$i \frac{d}{dt} \hat{A} = [\hat{A},\hat{H}].$$ Can we create a 'state' by ...
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1answer
64 views

Schrodinger equation contains two pieces of information?

I've read that Schrödinger equation contains two pieces of information: about amplitudes and about phases. I know continuity eqn. for probability which is nice expression of former. I wanted to ...
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2answers
127 views

Why is $ \psi = A \cos(kx) $ not an acceptable wave function for a particle in a box?

Why is $ \psi = A \cos(kx) $ not an acceptable wave function for a particle in a box with rigid walls at $x=0$ and $x=L$ where $$ k = \frac {(2mE)^{1/2}} {\hbar} \, ?$$ I had plugged the wave ...
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1answer
74 views

Width of a 1 dimensional box with same ground state energy as hydrogen atom [closed]

I am trying to find the width $L$ of a one-dimensional box for which the ground state energy of an electron in the box equals the absolute value of the ground state of a hydrogen atom. No ...
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2answers
67 views

How can a solution of the time-independent Schrödinger equation evolve in space?

I understand that if the Hamiltonian does not depend on the time, the Schrödinger Equation becomes separable, so you get $$ H \psi(x) = E \psi(x) $$ and $$ \Psi(x,t) = ...
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1answer
36 views

Potential energy function for high energy continuum?

For the hydrogen atom the quantised energy levels are: $$E_n = \frac{- 13.6 eV}{n^2}\quad\text{with}\quad n = 1,2,3...$$ One peculiar property of this quantisation is that for large $n$ the energy ...
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1answer
86 views

Derivation of time-dependent Schrödinger equation from De Broglie hypothesis

In a quantum mechanics script I'm reading, the Schrödinger equation is "derived" (more precisely, motivated) by the De Broglie hypothesis. It starts at $$ \lambda = \frac{2\pi h}{p} $$ $$ \omega = ...
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2answers
77 views

How to predict bound states in a 1 D triangular well?

Assume we have a (single) particle in a potential well of the following shape: For $x \leq 0$, $V = \infty$ (Region I) For $x \geq L$, $V = 0$ (Region III) For the interval $x > 0$ to $x < ...
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1answer
81 views

A particle in a 1D box: what is the meaning of velocity?

In the box $x = 0$ to $x = L$, $V = 0$, and for $x < 0$ and $x > L$, $V = \infty$ (infinite potential well). The eigenvalues of the Hamiltonian are: $$E_n = \frac{n^2 h^2}{8L^2} \, .$$ Since ...
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1answer
37 views

Problem with the Cooley-Numerov Method for Solving the Radial Nuclear Schodinger Equation in the Born-Oppenheimer Approximation

I have been trying to implement a solver for the radial nuclear Schodinger equation in the Born-Oppenheimer approximation using a similar method to R. J. Le Roy's LEVEL program[1]. I have as input a ...
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1answer
82 views

Schrodinger's equation with negative sign

In time dependent Schrodinger's equation as given in Schrodinger's lecture (Four Lectures on Wave mechanics, Blackie & Son, 1949, pg22) he arrives at $$\nabla^2\psi-\frac{4 \pi m ...
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2answers
81 views

Quantum electron and field interactions

What is the proper way to consider the electric field generated by an electron wavefunction governed by the Schrodinger equation? Can you get a result that would match observation, or is this a ...
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1answer
100 views

Why electron can not be found at some node locations in the infinite potential well?

Consider electron in an infinite potential well, studied in quantum mechanics. Position probability density of the electron is $$ P_n(x)=\left(\frac{2}{L}\right)\sin^2\left(\frac{n\pi x}{L}\right)$$ ...
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74 views

Where does the factor of $x$ come from in this formula for expectation value?

Given the normalised ground-state wave-function: $$\Psi(x, t)=\begin{cases} \sqrt\frac{2}{d}\cos(\frac{\pi x}{d})e^\frac{-i\hbar\pi^2t}{2md^2} & \ \lvert x\rvert<\frac{d}{2}, \\ 0 & ...
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28 views

Solutions to time-independent Schrödinger's equation with symmetrical (even) potential [duplicate]

A problem from Griffith's Introduction to Quantum Mechanics asks to prove the following: Given a symmetric potential $V(x)$ $(=V(-x))$, the solutions to the time-independent Schrödinger's equation ...
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3answers
221 views

Constructing solutions to the time-dependent Schrödinger's equation

The following question is from David Griffiths' Introduction to Quantum Mechanics: Problem 2.13 A particle in the harmonic oscillator potential starts out in the state $$\Psi(x,0) = A[3 ...
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1answer
63 views

Solution of the Radial Part of the Schroedinger Equation [closed]

The general Schroedinger Equation is: $$\left[-\frac{\hbar^2}{2m}\triangle +V(r,\vartheta,\varphi)\right]\psi_{nlm}=E\psi_{nlm}$$ When considering free waves, i.e. $V(r,\vartheta,\varphi)=0$ and a ...
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1answer
51 views

Property of the wave functions of a free particle

How can I show that the following holds? $$\langle nlm\mid \partial_z^2\mid nlm\rangle=-\int_0^{4\pi}d\Omega\int_0^{\infty}drr^2\left|\partial_z\psi_{nlm}\right|^2$$ The wave functions of a free ...
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1answer
80 views

Non-relativistic limit of complex scalar field Lagrangian

I am trying to derive the non-relativistic Lagrangian for a complex scalar field from taking the non-relativistic limit of the complex scalar field Lagrangian. I am following the steps in "QFT for ...
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1answer
46 views

Analytical non-separable solution for schrodinger equation

I am an undergraduate with the background of a first course in Quantum Mechanics. I want to find out if there exist non-unique solutions to Schrodinger equation. So I have to find potentials ...
0
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1answer
60 views

The Tunnelling Man [closed]

What's the probability that I will tunnel through a solid wall? By "tunnel" I mean, the probability of finding me on the other side of the wall. Assumptions Wall thickness = $d$ Clearly state any ...
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70 views

Selecting physical solutions in numerical eigenvalue problems

I try to solve a certain time-independent Schrodinger equation numerically, using the method of finite differences. My boundary conditions are such that the finite difference method gives me an ...
1
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1answer
68 views

Finding the wave function of a quantum harmonic oscillator [duplicate]

How can I find the wave function of a quantum harmonic oscillator? If I measure its energy several times, my measurements will change the state of a system. All I know are the possible states, given ...
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1answer
45 views

Particle in a box - speed probability distribution

Consider a particle in a box with infinite barriers. By solving the Schrödinger we can find the probability of finding the particle at some points in the box. How can we find the probability of ...
2
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1answer
82 views

Obtaining quantum Hamiltonian for charged particle from path integral formulation

I was working on Shankar 8.6.4, which is about obtaining in one dimension the Hamiltonian operator of a charged particle from the path integral formulation. First, I get the propagator over a time ...
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26 views

How can we derive the schrodinger wave equation? [duplicate]

How can we derive the Schrodinger wave equation with simplified explanation?
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1answer
125 views

Why “textbook examples” of solutions to Schrodinger equation only deal with electrons?

Whenever studying first courses of quantum mechanics, the Schrodinger eqaution is always illustrated by an electron in some kind of a potential, and the solution (wavefunction) represents probability. ...
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1answer
140 views

How did Schrödinger come up with his equation? [duplicate]

So all of the people who studied QM know the famous Schrödinger equation. I have read that it was not derived, but it is a postulate; something that is just real. Some people have tried to explain ...
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63 views

Intuition behind the solutions of the Schrodinger equations

I am learning the basic and easy first examples of the most common problems in Quantum Mechanics,and while trying to find solution to the Schrodinger equation,i find myself struggling with the ...
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45 views

Potential in Schrödinger equation when doing a Galilean transformation

I was looking at the quantum mechanics book by Bransden and Joachain, specifically at the section about Galilean transformations, and I was trying to find out what they did here for the potential ...
3
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2answers
131 views

Time evolution of a wavepacket

I do not understand why if $H\psi = E\psi$, then the time-evolution of the wavefunction is given by $e^{-iEt/h}\psi(x)$.
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1answer
55 views

Time dependent solution to infinite well

A particle of mass $m$ is confined within an infinite, one-dimensional potential well, $U(x)$, of width $a$. $$ U\left(x\right) = \left\{ \begin{array}{lr} \infty &\: x \leq ...
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3answers
65 views

Calculating the probability of a given energy

Given a normalised wavefunction say $$\psi(x) = A\sin(n\pi x),$$ (where $A$ is a normalisation constant) I can calculate the probability of finding the particle being between a position $x$ and $x + ...
2
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4answers
205 views

Physical reason why the derivative of a wavefunction has to be continuous?

Question What is the physical reason (i.e. without any maths) that the derivative of a wavefunction (except with infinite potentials) has to be continuous? Other info I know that in the classical ...
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2answers
52 views

Potential step and exponential decay?

Let us say we have a wave going from a region ($x<0$) where the potential is $U_1$ to a region ($x>0$) where the potential is $U_2$. The wave function in the second region takes the form: ...
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71 views

Naive quantization of Schrödinger field

I just started learning QFT and I was wondering if one is able to quantize the Schrödinger field similar to the way one is able to quantize electromagnetic or elastic mechanical wave modes. E.g. ...
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26 views

Metastable $E=0$ s-wave bound state in a spherical potential well

I am currently dealing with scattering theory. I looked up the scattering on a spherical well potential. $$V(r) = \begin{cases} -V_0 & , r \leq R\\ 0 & ,r > R \end{cases} $$ where ...
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1answer
82 views

Why is the ground state energy of a 2DEG higher compared to the 3DEG?

I am reading something about a 2DEG (2-dimensional electrongas model) and can not understand it. My book says the ground state of the 2DEG is higher compared to a 3DEG because the confinement to 2D ...
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1answer
53 views

What does it mean for a quantum particle to have energy $E_n$? And what is its general normalised state?

In this particular case, I have found the energy to be quantised with energy levels $\frac{h^2n^2}{2m} >0 $ where $n$ is an integer. Suppose a particle has energy $E=\frac{4h^2}{2m}$, then this ...
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1answer
74 views

Boundary conditions of the radial Schrodinger equation

Consider the radial differential equation $$\bigg( - \frac{d^2}{dr^2} + \frac{(\ell+\frac{d-3}{2})(\ell+\frac{d-1}{2})}{r^2} + V(r) + m^2 \bigg) \phi_\ell (r) = \lambda\ \phi_\ell (r),$$ which I've ...
2
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1answer
50 views

Normalizing a wave function in a mixed well

So I got this potential and want to solve for the even wavefunctions http://imgur.com/GKAy4nD Since it's symmetric around the origin I need only to look at the interval [0,b] and solve for the ...
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23 views

Ground State energy as a function of $N$ and $B$, $E_0(N,B)$

The one-particle Hamiltonian is given by: $$\hat{H}=\frac{1}{2m}\left(p+\frac{e}{c}A\right)^2$$ where $p=\hbar\vec{k}$ with $e > 0$ and vector potential $A=(0,x,0)B$, such $B=\triangledown ...