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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92 views
How to tell if a complex exponential blows up
I'm following Griffiths' Introduction to Quantum Mechanics, where he's discussing the general solution to the delta-function potential problem. The solution he refers to is ...
3
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3answers
352 views
Can we have discontinuous wavefunctions in the Infinite Square well?
The energy eigenstates of the infinite square well problem look like the Fourier basis of L2 on the interval of the well. So then we should be able to for example make square waves that are an ...
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1answer
73 views
Tunneling and transmission
Lets say we have a tunelling problem in the picture, where $W_p$ is a finite potential step:
If particle is comming from the left a general solutions to the Schrödinger equations for sepparate ...
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1answer
321 views
Even and Odd States of a 1D finite potential well
Is it possible for a particle trapped in a 1D finite potential well to evolve from a even state to an odd state and vice-versa? Why?
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2answers
204 views
Many-worlds: how often is the split how many are the universes? (And how do you model this mathematically.)
When I read descriptions of the many-worlds interpretation of quantum mechanics, they say things like "every possible outcome of every event defines or exists in its own history or world", but is this ...
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1answer
226 views
The eigenvalue of Schrodinger Equation
I'm a student majoring in Mathematics.But now I'm studying the KDV equation which uses Schrodinger Equation. My question is that in time-independent Schrodinger ...
3
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1answer
191 views
Projection of states after measurement
Continuing from the my previous 2-state system problem, I am told that the observable corresponding to the linear operator $\hat{L}$ is measured and we get the +1 state. Then it asks for the ...
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1answer
330 views
Finding $\psi(x,t)$ for a free particle starting from a Gaussian wave profile $\psi(x)$
Consider a free-particle with a Gaussian wavefunction,
$$\psi(x)~=~\left(\frac{a}{\pi}\right)^{1/4}e^{-\frac12a x^2},$$
find $\psi(x,t)$.
The wavefunction is already normalized, so the next thing to ...
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1answer
299 views
Can we solve the particle in an infinite well in QM using creation and annihilation operators?
The particle in an infinite potential well in QM is usually solved by easily solving Schrodinger differential equation. On the other hand particle in the harmonic oscillator oscillator potential can ...
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79 views
Change of basis in non-linear Schrodinger equation
At the mean-field level, the dynamics of a polariton condensate can be described by a type of nonlinear Schrodinger equation (Gross-Pitaevskii-type), for a classical (complex-number) wavefunction ...
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2answers
276 views
Plotting $\psi$ for finite square well potential
Lets say we have a finite square potential well like below:
This well has a $\psi$ which we can combine with $\psi_I$, $\psi_{II}$ and $\psi_{III}$. I have been playing around and got expressions ...
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3answers
92 views
How do I determine the location of a free particle with Schrödinger's equation?
I'm trying to get to grips with the Schrödinger equation by looking at a free particle. I'm certain at some point I massively misunderstood something.
According to a textbook and a lecture the free ...
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3answers
220 views
Is this interpretation of $\psi=\frac{1}{\sqrt{\pi a^{3}}}e^{-r/a}$ correct?
Apologies if this is stating the obvious, but I'm a non-physicist trying to understand Griffiths' discussion of the hydrogen atom in chapter 4 of Introduction to Quantum Mechanics. The wave equation ...
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2answers
890 views
Sinusoidal vs exponential wave functions with Schrodinger's equation
When solving Schrodinger's equation, we end up with the following differential equation:
$$\frac{{d}^{2}\psi}{dx^2} = -\frac{2m(E - V)}{\hbar}\psi$$
As I understand it, the next step is to guess the ...
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1answer
753 views
Probability current
Conservation of probability: Suppose a wavefunction has ${\partial \mathbb P \over \partial t} = -t f(x,t)$ and ${\partial j \over \partial x} = i f(x,t)$. How does it follow that ${\partial \mathbb P ...
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1answer
303 views
Solving time dependent Schrodinger equation in matrix form
If we have a Hilbert space of $\mathbb{C}^3$ so that a wave function is a 3-component column vector
$$\psi_t=(\psi_1(t),\psi_2(t),\psi_3(t))$$
With Hamiltonian $H$ given by
$$H=\hbar\omega
...
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1answer
234 views
Wavefunction in quantum mechanics and locality
Every wavefunction of a form $\Psi(x)$ can be described as a superposition of multiple free particle solutions.
We can see the following Fourier transform:
$$ \psi(x) = \int e^{ik\cdot x} \psi(k) dk ...
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4answers
552 views
solution of schrodinger equation - infinite solutions?
In Griffiths's introductory quantum mechanics book, it states that if $\Psi (x,t)$ is a solution to the Schrodinger's equation, then $A\Psi (x,t)$ must also be a solution, where A is any complex ...
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2answers
305 views
Correct application of Laplacian Operator
Not a physicist, and I'm having trouble understanding how to apply the Laplacian-like operator described in this paper and the original. We let:
$$ \hat{f}(x) = f(x) + \frac{\int H(x,y)\psi(y) ...
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1answer
262 views
Superposition of wavefunctions
Suppose you have 2 normalized wavefunctions $\psi_1=Ne^{iax}e^{if(x)}e^{i\omega t}$ and $\psi_2=Ne^{-iax}e^{if(x)}e^{i\omega t}$ defined on $x\in [-x_0,x_0]$ and vanishes for $|x|>x_0$. What then ...
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1answer
2k views
How to solve this Schrödinger equation?
I am taking an intro level quantum mechanics class. Our textbook gives a problem like this:
The deuteron is a nucleus of "heavy hydrogen" consisting of one proton and one neutron. As a simple ...
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3answers
102 views
When does the time independent Schrödinger equation have a physical solution?
In some cases, such as finite and infinite square wells, the Hamiltonian has energy eigenstates which correspond to physical wavefunctions.
In other cases, such as a one dimensional universe with ...
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1answer
134 views
Usage of Schrödinger equation vs Madelung equations
It is well known that Madelung formulation is alternative to the Schrödinger Formulation, cf. this previous Madelung transformation Phys.SE post. I wanted to know what makes Schrödinger's formulation ...
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2answers
402 views
Two expressions for expectation value of energy
I was looking up expectation value of energy for a free particle on the following webpage:
http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/expect.html
It says that $E=\frac{p^2}{2m}$ and ...
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1answer
292 views
Derivation of Bloch's theorem
I'm having a problem following a derivation of Bloch's theorem, looking at a one dimensional lattice with $N$ nodes and spacing a, we impose periodic boundary conditions, meaning that the ...
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2answers
187 views
Classical limit of a quantum system
If we have a one dimensional system where the potential
$$V~=~\begin{cases}\infty & |x|\geq d, \\ a\delta(x) &|x|<d, \end{cases}$$
where $a,d >0$ are positive constants, what then is ...
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1answer
799 views
How to solve Schrodinger Equation - Tunnelling
I have to solve analitically the Schrodinger equation in one-dimension with a barrier of potential (tunnel effect):
$$ih \frac{d}{dt} U(x,t) = \left[ \left(-h^2 \frac{d^2}{dx^2} \right) + q V(x) ...
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1answer
758 views
How to calculate time evolution of a wave function in an 1D infinite square well potential?
A particle in an infinite square well has an initial wavefunction
$$\psi (x,0) ~=~ Ax(a-x) \qquad \mathrm{for}\qquad 0\leq x\leq a.$$
Now the question is to calculate $\psi (x,t)$.
I have ...
2
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1answer
161 views
Degeneracy and the Hamiltonian
How many linearly independent eigenfunctions can be associated with one degenerate eigenvalue of the Hamiltonian operator? (Is there a limit since it contains a 2nd order differential operator?) ...
2
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1answer
1k views
Degeneracies of the first excited state
I have a box with $x,y,z$ all ranging from 0 to $l$. It has $V(x)$=0 inside and =$\infty$ outside. By extending the 1D Schrodinger equation, I have that the allowed energy eigenvalues are ...
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2answers
93 views
Why the hydrogen radial wave function is real?
Why the hydrogen radial wave function is real?
Is it a coincidence?
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1answer
357 views
Schrödinger equation with complex potential
In 1 dimension what is the solution of the Schrödinger equation with potential
$$ V(x) = V_r + i V_i $$
Potentials are constant.
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2answers
429 views
What math is needed to understand the Schrödinger equation?
If I now see the Schrödinger equation, I just see a bunch of weird symbols, but I want to know what it actually means. So I'm taking a course of Linear Algebra and I'm planning on starting with PDE's ...
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1answer
217 views
Explanation of equation that shows a failed approach to relativize Schrodinger equation
I'm reading the Wikipedia page for the Dirac equation:
$\rho=\phi^*\phi\,$
......
$J = -\frac{i\hbar}{2m}(\phi^*\nabla\phi - \phi\nabla\phi^*)$
with the conservation of probability ...
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2answers
124 views
can we apply WKB method for curved space times
let be the Hamiltonian of a surface $ H= g_{a,b} p^{a}p^{b} $ (Einstein summation assumed) my question is if although the space time is curved then can we use the WKB approximation to get the quantum ...
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1answer
517 views
Calculating Ground State Energy in 1D Potential
Given potential $V(x) = Asec(x)$ for $x > 0$. I want to calculate the ground-state energy $E_0$ via the Schrödinger equation.
I'm completely stuck on this one. I've set up the time-independent ...
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1answer
371 views
How far can you get (in quantum mechanics) with just commutation relations?
Clearly it is possible to derive a set of commutation relations from some Hamiltonian, and certainly they give useful and interesting invariants when investigating the behavior of quantum systems. ...
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1answer
1k views
Bound States in a Double Delta Function Potential [closed]
Let $V(x) = −u \delta(x) - v \delta(x − a)$ where $u, v > 0$ correspond to a potential with two $\delta$ wells. Let $v > u$. If $a$ is very large, there is certainly a bound state: the particle ...
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1answer
278 views
Is it possible that Atomic Electron Probability Density is a result of Heat?
The Schrödinger Equation provides a Probability Density map of the atom. In light of that, are either of the following possible:
The orbital/electron cloud converges to a 2d surface without heat ...
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2answers
87 views
What is the spectrum of energies for the potential $ a^{x} $?
Given a certain potential $ a^{x} $ with positive non-zero 'a' are there a discrete spectrum of energy state for the Schrodinger equation
$$ \frac{- \hbar ^{2}}{2m} ...
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1answer
255 views
Is it a total or an explicite time derivative in the Schrödinger equation?
I am always dubious when I need write Schrödinger equation: do I write $\partial / \partial t$ or $d/dt$ ?
I suppose it depends on the space in which it is considered. How?
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3answers
665 views
Analytic solutions to time-dependent Schrödinger equation
Are there analytic solutions to the time-Dependent Schrödinger equation, or is the equation too non-linear to solve non-numerically?
Specifically - are there solutions to time-Dependent Schrödinger ...
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2answers
336 views
Schrödinger's equation, time reversal, negative energy and antimatter
You know how there are no antiparticles for the Schrödinger equation, I've been pushing around the equation and have found a solution that seems to indicate there are - I've probably missed something ...
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2answers
492 views
Barrier in an infinite double well
I am stuck on a QM homework problem. The setup is this:
(To be clear, the potential in the left and rightmost regions is $0$ while the potential in the center region is $V_0$, and the wavefunction ...
1
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1answer
268 views
The Hermiticity of the Laplacian (and other operators)
Is the Laplacian operator, $\nabla^{2}$, a Hermitian operator?
Alternatively: is the matrix representation of the Laplacian Hermitian?
i.e.
$$\langle \nabla^{2} x | y \rangle = \langle x | ...
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1answer
238 views
Solving Schrödinger's equation for a specific potential
I am trying to solve this differential equation:
$$-\chi''(\epsilon)+\Big[\epsilon^2+\frac{2F}{hw}\sqrt{\frac{h}{hw}}\epsilon \Big]\chi(\epsilon)=\mu\chi(\epsilon) \tag1$$
This was found ...
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1answer
239 views
A quantum particle in a box (with a catch)
I am reading Shankar's Quantum Mechanics and I am looking at the case where there is one particle inside a box, where the potential is zero inside the wall and abruptly goes to infinity outside the ...
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2answers
223 views
Schematic expression of the Schrodinger equation
it would be great if someone could help me understand the following quote regarding wavefunctions :)
"$$\psi(x)=\sum_n C_nu_n(x)+\int dE C(E)u_E(x)$$ The expression is schematic because we have ...
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1answer
85 views
The Klein–Gordon equation
As we know that the Schrödinger equation presents basis of Quantum Mechanics and analogy with Newton second law in Classical Mechanics, I thought that relativistic interpretation of Schrödinger ...
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1answer
140 views
Finite, square, potential well
Lets say we have a finite square well symetric around $y$ axis (picture below).
I know how and why general solutions to the second order ODE (stationary Schrödinger equation) are as follows for ...
