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29 views

Dyson Series Iteration - Gives Exact Solution?

When we derive the Dyson series for usage as the time evolution operator in the case of a time dependent Hamiltonian, we start with the equation: \begin{align}\hat{U}_I(t,t_i) = 1 - \frac{i}{\hbar}\...
0
votes
1answer
27 views

Probability density of time-dependent wave functions

Why is it so that probability density of eigenfunctions of time-dependent schrodinger equation are time independent while that of general wave functions (which are a combination of the eigenfunctions) ...
9
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2answers
699 views

Do atomic orbitals “pulse” in time?

I understand that atomic orbitals are solutions to the time-independent Schrödinger equation, and that they are are analogous to standing waves ("stationary states"). However, even a standing ...
0
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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 ...
0
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0answers
41 views

Discrete time evolution in a non-Euclidean space?

The time independent schrödinger equation can be written as $$i\frac{\partial \psi}{ \partial t}=H\psi$$ if we consider the case of a 1D particle we can evolve it in time by discretising the ...
1
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1answer
96 views

Wave function evolution of an electron [closed]

In many basic quantum mechanics books the wave packet of an electron is described. It will say that the wave packet will broaden as time evolves because of dispersion. But suppose the electron just ...
2
votes
4answers
118 views

Why do wavefunctions for stationary states include $e^{-iEt/\hbar}$? [duplicate]

Stationary states are separable solutions with $\Psi(x, t)=\psi(x)e^{-iEt/\hbar}$. But why is that there? Griffiths (Section 2.1 Stationary states, equation 2.8) says that observables for these states ...
2
votes
4answers
230 views

Does the Schrodinger wave function associated with a non-moving free particle change in time?

I'm a bit confused by an answer given on this question. In the answer with the animation of a moving free (chargeless) particle and a non-moving free particle (or a free particle with a non-zero ...
-2
votes
1answer
129 views

Time evolution of a free particle with a given initial state [closed]

My homework problem reads: Consider a free particle in one dimension. Write an expression for the wavefunction $\psi(x, t)$ given an initial state $\psi_0(x) = Ae^{-ax^2}$ at $t = 0$, where $A$ is ...
0
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1answer
63 views

“General” for time evolution of quantum state

I am reading a book in which at some point they find the time-evolved wavefunction $\phi_0(\mathbf{r},t)$ from the static $\phi_0(\mathbf{r})$. They say that "employing the Heisenberg time evolution ...
2
votes
2answers
222 views

What's the time derivative of the Annihilation operator?

I've been dealing with annihilation operator recently where you can see related information Time derivative of the state vector as expressed in abstract Hilbert space vs. as a wavefunction How to ...
0
votes
1answer
72 views

What is the state of particle at time $t$ if at $t=0$ it is in an eigenstate of $\hat{A}$, and $\hat{A}$ commutes with $\hat{H}$?

EDIT: added (assuming $\lambda$ to be non-degenerate). Based on the specifics of the question, we don't in fact know whether this is the case, so it may be that $\left|\lambda\right>$ is not an ...
0
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1answer
143 views

Solving the Schrodinger equation with a time-dependent Hamiltonian

I am trying to find the general solution to the Schrodinger equation with a time-dependent Hamiltonian: $$ i \frac{\partial}{\partial t}| \psi(t) \rangle = H(t) | \psi(t) \rangle.$$ My Hamiltonian ...
2
votes
4answers
848 views

How do base kets satisfy Schrödinger's equation in Schrödinger picture and why don't they evolve with time?

According to Sakurai, eigenvalue equation for an operator $A$, $A|a'\rangle=a'|a'\rangle$. In the Schrödinger picture, $A$ does not change, so the base kets, obtained as the solutions to this ...
2
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0answers
146 views

Infinite square well and Heisenberg picture

The infinite square well is often a mainstay of introductory quantum physics courses. Its boundary conditions at the well-walls are easily solved to the find the Hamiltonian's eigenfunctions in the ...
2
votes
2answers
192 views

Is there a unitary transformation such that the Hamiltonian in the time-dependent Schrödinger equation becomes real symmetric?

The time-dependent Schroödinger equation is given as (with $\hbar=1$): $$i\dfrac{d}{dt}\psi(t)=H(t)\psi(t)\ ,$$ where $\psi$ is some normalized column vector and $H(t)$ is a Hermitian matrix with time-...
0
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0answers
103 views

Definition of Hamiltonian in Quantum Mechanics [duplicate]

Is there any particular reason that the Hamiltonian operator was defined in quantum mechanics to be $$\hat H := \frac{\hat p^2}{2m} + V$$ as opposed to $$\hat H := i\hbar \frac{\partial}{\partial t}?$$...
4
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3answers
267 views

Can a second-order Schrödinger equation preserve the norm?

Suppose we lived in a universe in which the Schrödinger equation contains second order time derivatives, $$i\hbar \partial_t^2|\varphi(t)\rangle = \mathbb{H} | \varphi(t)\rangle.$$ Would it be true ...
1
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1answer
75 views

Is it an assumption or truth for spatial and temporal variables separation if Hamiltonian is time-indepdent in Schrodinger equation?

For the derivation from time-depdent Schrodinger equation to time-indepdent Schrodinger equation, if the Hamiltonian is time-indepdent, we assume the spatial and temporal variables in the wave-...
5
votes
2answers
207 views

Is there an equivalent to the Schrodinger equation for quantum mechanics over the reals?

Many people have considered alternatives to standard quantum mechanics in which the Hilbert space is over the real instead of the complex numbers - see e.g. here, here, here, here, here, and here. In ...
0
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1answer
77 views

A fundamental question about Time-dependent Hamiltonians

I have a fundamental question about Quantum Mechanics or even mechanics in general. I am aware that there are stationary solutions and non-stationary solutions. The stationary solutions solve ...
0
votes
1answer
189 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 ...
0
votes
1answer
42 views

State of a system at previous time

If I am given the state of a quantum system at $t=0$ as $| \psi \rangle$ and I know the Hamiltonian $H$ of the system for time $t<0$, how can I write the state of the system at some time $t<0$?
0
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0answers
26 views

Evolution of a quantum state due to a pulse

I have some Lorentzian pulse with equation $L(\omega)=\frac{1}{\pi}\frac{\frac{1}{2}\Gamma}{((\omega-\omega_0)^2+(\frac{1}{2}\Gamma)^2)}$ with Fourier transform $F(t)=e^{(-2\pi it\omega_0-\Gamma\...
2
votes
1answer
121 views

Are there 'closed' solution of Schroedinger equation?

Are there closed solution for the wave function of schroedinger equation? i mean solutions in the form $ \Psi (x,t)= f(x-t,y,z,t) $ that are not given by infinite series. For example for the 1+1D ...
-1
votes
3answers
249 views

Why does time evolution preserve the norm of a wavefunction?

I saw an awesome derivation of Schrodinger's equation on Wikipedia. Part of it relies on: Since time-evolution must preserve the norm of the wave-function, it follows that $U(t', t)$ must be a ...
1
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0answers
89 views

How to numerically implement a Wick rotation?

I'm solving a Schroedinger-type differential equation using numerical methods (RK4 for precision, explicit Euler to get a rough idea). I have an initial condition to start. I understand that replacing ...
7
votes
3answers
835 views

How is Schrödinger evolution experimentally verified?

This is surely a pretty ignorant question, but I'm just starting out learning physics. In quantum theory the evolution of a particle's wavefunction is supposed to have two stages: deterministic ...
0
votes
1answer
195 views

Connecting the Schrodinger equation, unitarity and norm-preserving of states in time evolution

This may be really basic but I'm having trouble connecting the following issues: 1) The 2-norm for state vectors is preserved during time evolution 2) The Hamiltonian is a Hermitian operator. With ...
2
votes
2answers
174 views

Ordinary vs. partial derivatives of kets and observables in Dirac formalism

I'm a bit confused as to when ordinary and partial derivatives are used in the Dirac formalism. In the Schrödinger equation, for instance, Griffiths [3.85] uses ordinary derivatives: $$ i \hbar \...
-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 $\psi_{present}(r)$. ...
3
votes
1answer
118 views

What's a reasonable 'translation' of the Schrödinger equation?

For this form of the equation: $$\hat H|\psi(t)\rangle = i \hbar \frac{\partial{}}{\partial{t}}|\psi(t)\rangle.$$ For instance: "The total energy of a quantum state at time t is equal to $i\hbar$...
4
votes
2answers
773 views

How can I prove that the wave function remains normalized as time goes?

Exploiting Schrödinger equation and its conjugate we can show that $$ \dot{\Psi} = \frac{i \hbar}{2m} \nabla^2 \Psi - \frac{i}{\hbar} U \Psi $$ $$ \dot{\Psi}^* = -\frac{i \hbar}{2m} \nabla^2 \Psi^* + \...
0
votes
3answers
95 views

Understanding the time evolution of a quantum state

I am trying to understand this equality below, but I can't seem to wrap my head around it. The Hamiltonian is defined as $\vec H=-\frac {\gamma B \hbar}{2}\sigma_x$, which gives the eigenvalues $\pm \...
10
votes
3answers
2k views

Why is the momentum-space wavefunction for a free particle not a function of time?

Suppose the initial wave function of a free particle is given by $\psi(x,0)$. Now to find how the wave function evolves with time we generally do the Fourier transform of the wave function at $t=0$. ...
1
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0answers
138 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
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1answer
391 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 ...
3
votes
3answers
917 views

Schrödinger equation for time dependent Hamiltonian and conjugation

The Schrödinger equation for the evolution operator reads: $$ \frac{\partial U}{\partial t} = -\frac{i}{\hbar}HU $$ where for a time dependent Hamiltonian which need not commute with itself at ...
1
vote
0answers
38 views

Time-evolution of localised particle [duplicate]

I am interested in the question of, if a particle is initially localised at some position $x_0$ what it will evolve to at a later time assuming a free Hamiltonian $H = p^2 /2m$. Long story short, I ...
3
votes
3answers
2k views

Why is time-evolution operator unitary?

When we shift the system's time from $t=0$ to $t = t$, we can define the following operator $\hat{U}$. $$\hat{U} = e^{- i \hat{H} t / \hbar} \, .\tag{1}$$ So many (as far as I read, almost all of) ...
0
votes
4answers
814 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 ...
0
votes
2answers
472 views

General solution of states of time dependent Hamiltonian

Given a time dependent Hamiltonian which commutes at different times, we have the time evolution operator given by $$\mathcal{U}(t,0) = \text{exp}\bigg[-\bigg(\frac{i}{h}\bigg)\int_{0}^{t}dt' H(t')\...
1
vote
2answers
503 views

How do I find the time evolution of a ket?

I have a question which reads: Let \begin{bmatrix} {E_0} & 0 & A \\ 0 & E_1 & 0 \\ A & 0 & E_0 \end{bmatrix} be the matrix representation of ...
1
vote
1answer
620 views

Time evolution of operators, derivation

I don't understand something about the Heisenberg and interaction picture, in my notes the time evolution of operators for the Heisenberg and interaction picture is derived, by inserting them into the ...
6
votes
3answers
615 views

What is a “picture” in quantum mechanics?

One of the basic ingredients of quantum mechanics is the possibility of working in different "pictures". Thus, while we normally work in the Schrödinger picture, in which states evolve according to ...
1
vote
3answers
159 views

Time-dependence in QM

When you have a system governed by some Hamiltonian H that starts in some state $\lvert a \rangle$, the way to calculate its time-dependence is to decompose into a linear combination of eigenstates of ...
0
votes
2answers
520 views

Phase factors of eigenstates for a time-dependent Hamiltonian

For a time-dependent Hamiltonian, the Schrödinger equation is given by $$i\hbar\frac{\partial}{\partial t}|\alpha;t\rangle=H(t)|\alpha;t\rangle,$$ where the physical time-dependent state $|\alpha;t\...
1
vote
2answers
2k views

Heisenberg equation with time-dependent Hamiltonian

It is the root of quantum mechanics that Heisenberg picture and Schrödinger picture are equivalent? In most textbooks and wikipedia, the equivalence is proved with a time-independent Hamiltonian. ...
0
votes
1answer
151 views

Time dependence of wave packets without eigenfunctions

In general, to obtain the time dependence of an arbitrary wave packet $\left| \phi(x)\right>$ in the Schödinger picture, we expand the wave packet in the energy eigenfunction basis $\left| \psi_n(x)...
1
vote
0answers
56 views

Exactly solvable quantum dynamics problems [duplicate]

By exactly solvable model, people generally mean models whose eigenstates or eigenenergies can be solved analytically. Simple examples are the harmonic oscillator, the infinitely deep square well ...