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2answers
49 views

Can the wave-function of any particle in any basis be written as a matrix?

Can the wave-function of any particle in any basis be written as a matrix? If no, how can we explain this, where the Hamiltonian $H$ in U is a QM operator that can be written as a linear ...
-1
votes
1answer
86 views

Quantum field theory , Schrödinger wavefunction

$\psi$ is a state that given $|\psi\rangle=\int d^3x\psi(x)|x\rangle$ How does the wave function change in time? The Hamiltonian can be written as $$H=\int d^3x\frac{1}{2m}\nabla\psi^*\nabla\psi=\int ...
0
votes
1answer
36 views

Can we automatically find the Hamiltonian from knowledge for multiple wave functions?

Say we had a set of wave functions $\psi(x,t)$ that we new the values of for all $x$ and $t=t_0..t_1$. Say we had $N$ of these wavefunctions, perhaps $N=10$. All these wave functions start off at ...
0
votes
0answers
79 views

Solution to two non-coupled quantum harmonic oscillators

Given the following Hamiltonian: $$\hat H = -\frac{\hbar^2}{2m}\frac{\partial^2}{\partial x_1^2} -\frac{\hbar^2}{2m}\frac{\partial^2}{\partial x_2^2} + \frac{1}{2}m\omega_1^2x_1^2 + \frac{1}{2}m\...
1
vote
1answer
55 views

Domain of the infinite square well hamiltonian

I am reading the book by Gitman et al. 'self-adjoint extensions in quantum mechanics'. In the book, they give a precise definition of the domain of the hamiltonian of an infinite square well. For ...
0
votes
2answers
41 views

Wave function of a particle under $V(x)$ (QM)

Suppose I have a particle with mass $m$ and it's under potential of a certain $V(x)$. (NOT an infinite or finite potential well) Also given is the wave function at time $t=0$, $\psi(x,0)$. What is ...
2
votes
2answers
63 views

What is the simplest possible Hamiltonian that yields an Antisymmetric Wavefunction?

I am using a Split-Operator Fourier Transform (SOFT) technique to solve the time-dependent electronic Schrödinger Equation (TDSE) for a Hydrogen molecule under the Born-Oppenheimer approximation. So I ...
0
votes
2answers
62 views

Two ways to define wave function in Heisenberg picture

I found two ways to define a wave function in Heisenberg picture, $| \psi(t) \rangle_\mathrm{H}=\mathrm e^{\mathrm i H t/\hbar} | \psi(t) \rangle_\mathrm{S}$ which further gives $|\psi(t) \rangle_\...
0
votes
1answer
49 views

False solution of Landau Hamiltonian

The Landau Hamiltonian in 2D is given (in natural units $q=c=2m=1$) by $$ \hat{H} = (\hat{\vec{p}}-\vec{A}(\hat{\vec{x}}))^2 \,,$$ where $\vec{A}$ is the magnetic vector potential field. We know that ...
2
votes
1answer
121 views

Solving Schrödinger equation by neural networks - trial function explanation

I'm reading this paper about solving Schrödinger equation using the combination of genetic algorithm and neural networks. But one part confuses me - the author defines his trial function, i.e. the ...
1
vote
1answer
35 views

Intro QM representation of Abraham-Lorentz Force

What does the Schrodinger equation look like if you add some term for the Abraham-Lorentz force? I get a self reference term I'm not sure how to handle. I realize this is probably addressed by QED, ...
0
votes
1answer
191 views

Integral of Schrodinger equation for a time-dependent Hamiltonian

I am given the following Hamiltonian, $H=H_1=\frac{p^2}{2m}+\frac{1}{2}m\omega_1^2x^2$ for $t<0$ and $H=H_2=\frac{p^2}{2m}+\frac{1}{2}m\omega_2^2x^2$ for $t\geq 0$. Now I want to integrate my ...
1
vote
0answers
157 views

One Hamilton Operator for two independent harmonic oscillators

If we consider two independent harmonic oscillators (identical too a two dimensional harmonic oscillator), the hamilton operator is $$ H = \frac{p_1^2}{2m_1}+\frac{1}{2}m_1\omega_1^2x_1^2 + \frac{...
5
votes
2answers
307 views

Expectation Value $\langle \frac{1}{r^2} \rangle$ using Hellmann–Feynman theorem

Suppose we have the hydrogen atom$$ H ~=~ \frac{p_r ^2}{2m} + \frac{L^2}{2mr^2} -\frac{e^2}{r} \,.$$And have solved the Schrodinger equation finding $$E_n = - \frac{me^4}{2 \hbar ^2 n^2} $$ and $$ Ψ_{...
1
vote
1answer
408 views

Why stationary state wave functions are eigenfunctions of the Hamiltonian operator?

can somebody please explain why stationary state wave functions are eigenfunctions of the Hamiltonian operator?
2
votes
3answers
137 views

Single electron on a two atoms chain : factorisation of hilbert space by external and internal degrees of motion

I have a question on the first pages of the book "A Short Course on Topological Insulators" by János K. Asbóth, László Oroszlány and András Pályi But actually we can see it here : http://theorie....
-1
votes
1answer
53 views

How to solve Schrodinger equation back in time to find past wavefunction from which present wavefunction has been evolved?

How to solve Schrödinger equation back in time to find past wavefunction from which present wavefunction has been evolved? i.e. Suppose, at present or at this moment I know $\psi_{present}(r)$. ...
0
votes
1answer
316 views

Average squared Hamiltonian of linear combination of eigenfunctions

As part of a larger problem, I am trying to find the average squared Hamiltonian of a system with eigenfunctions $\psi_{1,1}$, $\psi_{1,2}$, $\psi_{2,1}$, $\psi_{2,2}$ and the following wave function: ...
1
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0answers
139 views

Origin of the Schrodinger equation by L. D. Landau and E. M. Lifshitz

In the book "Quantum Mechanics" by L. D. Landau and E. M. Lifshitz, it is mentioned that, "The wave function Ψ completely determines the state of a physical system in quantum mechanics. This means ...
0
votes
1answer
396 views

Is it possible to study a time-dependent Hamiltonian in Schrödinger picture?

Operators in Heisenberg picture are time-dependent while those in Schrödinger picture are time-independent, and they are related by $$A_H(t)=U^\dagger(t,t_0)A_S(t_0)U(t,t_0)$$ where $U(t,t_0)$ is the ...
1
vote
2answers
423 views

Is every eigenfunction of angular momentum magnitude necessarily also an eigenfunction of total energy?

Is every eigenfunction of angular momentum magnitude necessarily also an eigenfunction of total energy? Is the reciprocal statement true? This question is from Eisberg and it sounds very confuse to ...
6
votes
2answers
510 views

How is an operator applied to a wavefunction in quantum mechanics? [closed]

If you have the Hamiltonian operator written as such: $$\hat H = -\frac{\hbar}{2m}\frac{1}{r}\frac{\partial^2}{\partial r^2}r \tag{1}$$ then to apply the Hamiltonian operator to a wavefunction, do ...
1
vote
0answers
148 views

Zero Energy Wavefunctions in 1D P-Wave Kitaev Model

$\bf{Setup}$ Hi! I am trying to derive the wavefunctions of the zero energy solutions of the Schrodinger equation in a 1D p-wave superconductor (Kitaev model). I am starting with the Hamiltonian $ \...
0
votes
4answers
816 views

Why do we search for stationary solutions to the Schrodinger equation for potential wells?

When considering potential wells textbooks simply say that we search for the stationary solutions of the schrodinger equation. Why do we do this? What tells us that the wavefunction will be ...
4
votes
4answers
2k views

Translationally invariant Hamiltonian and property of the energy eigenstates

If the Hamiltonian of a quantum mechanical system is invariant under spatial translation, then the linear momentum is a constant of motion. Apart from that, can we make some comment about the nature ...
1
vote
2answers
142 views

Wave function in a semi infinite line [closed]

How to normalize the wave function of a particle in a semi-infinite 1D interval $x\in [0, \infty],$ with boundary condition $\varphi(0)=0\,?$ Hamiltonian is $H=P^2/2M$ and wave function $\varphi(x)=C\...
6
votes
4answers
829 views

Does the Hilbert space include states that are not solutions of the Hamiltonian?

I've studied Quantum Mechanics and I know the usual answer "The dimension of the Hilbert Space is the maximum number of linear independent states the system can be found in". There is something about ...
3
votes
1answer
3k views

Expectation value of Hamiltonian?

Consider a particle in an infinite potential well with length $L$: $\forall x \in (0,L): V(x) = 0$ and $V(x) = \infty$ elsewhere. The wave function at time $t = 0$ is given by $$ \psi(x,0) = \begin{...
15
votes
4answers
1k views

Is it possible to reconstruct the Hamiltonian from knowledge of its ground state wave function?

Is it possible to "construct" the Hamiltonian of a system if its ground state wave function (or functional) is known? I understand one should not expect this to be generically true since the ...
1
vote
0answers
212 views

How to plot numerically the wave functions according to the Hamiltonian?

It is often difficult to analytically solve the Schrodinger equation, and so we need to obtain a solution numerically. An example plot is shown below. Here, the wave functions for a three junction ...
0
votes
1answer
468 views

Does the Hamiltonian time-evolution operator actually change the state of the system?

According to my understanding of things, the time evolution operator in QM looks something like this, $$U = \exp(-iHt/\hbar)$$ Which acts on the state vector / wave-function of the system to ...
1
vote
0answers
305 views

Expectation value of the Hamiltonian [closed]

How to calculate expectation value of the Hamiltonian for hydrogen atom? $$\langle H \rangle_{\alpha l} \equiv \frac{\langle \psi_{\alpha l m}|H(r)| \psi_{\alpha l m}\rangle} {\langle \psi_{\alpha l ...
0
votes
1answer
98 views

Energy of hydrogen atom - Schrodinger equation [closed]

The wavefunction of the electron in the hydrogen atom is $ k* exp(-r/a)$ (k is the normalization constant), but it does not take n into account, whereas the solution of Schrödinger's equation ($H(...
1
vote
1answer
107 views

Can we measure the energy of one of several identical particles?

Suppose we have a many-particle system described via a many-particle wavefunction that involves single-particle states $\lvert\lambda_{a}\rangle$, $\lvert\lambda_{b}\rangle$, $\lvert\lambda_{c}\rangle$...
0
votes
1answer
239 views

Energy Expectation Value

I had an assignment question in which I was asked to calculate the expectation value of energy, $\langle E\rangle (t),$ and in the solution to it, the following was stated: \begin{align*} \langle E\...
3
votes
1answer
697 views

What happens to the wave function of a particle immediately after measuring its energy?

For this question, I will be adhering to the Copenhagen interpretation (since that's what I've learned in university so far). For the sake of brevity/clarity, also, assume the Hamiltonian here has ...
1
vote
2answers
243 views

Calculation of the $\langle H \rangle$ for a particle in a box

I am working through a problem in which a particle is in an infinite potential well of length $L$ at $t=0$ before the spontaneous change of the box being expanded to length $2L$. I have calculated the ...
1
vote
0answers
121 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 $$H=-\sum_{i=1}^3\frac{\partial^2}{\partial x_i^2}+\sum_{1\leq j<k\leq 3}2c\delta(x_j-x_k).$$...
1
vote
0answers
169 views

What happens to the Hamiltonian of the wave function after measurement?

As I understand it, the Hamiltonian is the kinetic plus the potential energy of the wave function. When a measurement is done what happens to the kinetic and potential energy? Does it dissipate? Is ...
1
vote
0answers
86 views

What is the physical significance of the two integration constants that appear in the ladder operator decomposition of the Quantum Hamiltonian?

If I have a simple one dimensional Hamiltonian of the form \begin{align} H = V - \partial_x^2 \end{align} and if I know one zero energy state solution $H\psi_0=0$ then I can use the Wronskian to ...
2
votes
0answers
344 views

Fermion 1D Hubbard Model ground state in the U = 0 limit

I am trying to determine the ground state of the 1D fermionic Hubbard model at half-filling of $2L$ sites with $L$ electrons with spin-$\uparrow$ and $L$ electrons with spin-$\downarrow$ in the $U=0$ ...
1
vote
2answers
2k views

Variational Theorem proof

I have been trying to prove variational theorem in quantum mechanics for a couple of days but I can't understand the logic behind certain steps. Here is what I have so far: \begin{equation} E=\frac{\...
2
votes
1answer
249 views

On use of Hamiltonians for Helium

The Hamiltonian of helium can be expressed as the sum of two hydrogen Hamiltonians and that of the Coulomb interaction of two electrons. $$\hat H = \hat H_1 + \hat H_2 + \hat H_{1,2}.$$ The wave ...
2
votes
1answer
324 views

Expectation value of Hamiltonian in different pictures of quantum mechanics

We start with the familiar Schrodinger equation: $$ i\hbar \frac{\partial \left|\psi_S\right\rangle}{\partial t} = \hat{H}_S \left|\psi_S\right\rangle $$ As we switch to a different picture than ...
2
votes
1answer
20k views

Calculating the expectation value of a Hamiltonian

I want to calculate the expectation value of a Hamiltonian. I have a wave function that is $$\psi = \frac{1}{\sqrt{5}}(1\phi_1 + 2\phi_2).$$ I want to know if I set this up properly. The Hamiltonian ...
-1
votes
1answer
201 views

Apply the Heisenberg Equation to the Hamiltonian [closed]

$\frac{d}{dt}$$\hat{H}$ = $\frac{i}{\hbar}$$[\hat{H},\hat{H}]$ +$\frac{\partial{\hat{H}}}{\partial{t}}$ That's as far as I've got. I do not know much about the Heisenberg equation or even what it ...
24
votes
2answers
2k views

The formal solution of the time-dependent Schrödinger equation

Consider the time-dependent Schrödinger equation (or some equation in Schrödinger form) written down as $$ \tag 1 i \partial_{0} \Psi ~=~ \hat{ H}~ \Psi . $$ Usually, one likes to write that it has a ...
2
votes
1answer
113 views

What does $\psi_j(r_i)$ mean?

I have a mean-field Hamiltonian for N electrons. The mean-field potential felt by electron $i$ at position ${\bf r}_i$ is given by $V^{(i)}_{int}({\bf r}_i)=\sum_{j\ne i}|\psi_j({\bf r}_i)|^2$ I ...