What is Stationary states: Definition and 34 Discussions
A stationary state is a quantum state with all observables independent of time. It is an eigenvector of the Hamiltonian. This corresponds to a state with a single definite energy (instead of a quantum superposition of different energies). It is also called energy eigenvector, energy eigenstate, energy eigenfunction, or energy eigenket. It is very similar to the concept of atomic orbital and molecular orbital in chemistry, with some slight differences explained below.
For a state to be stationary it must be time independent.
Naively, I tried to find the values of c where I don't have any time dependency.
##e^{c \cdot L_z} \psi (r,t) = e^{c L_z} \sqrt{\frac{8}{l^3}} sin(\frac{2 \pi x}{l}) sin(\frac{2 \pi u}{l}) sin(\frac{2 \pi z}{l}) e^{-iEt/\hbar}##...
Hi,
I have hard time to really understand what's a stationary state for a wave function.
I know in a stationary state all observables are independent of time, but is the energy fix?
Is the particle has some momentum?
If a wave function oscillates between multiple energies does it means that the...
Here we are talking about non-relativistic quantum physics. So we all know kinetic energy T = E - V = \frac{1}{2}mv^2 in classical physics. Here V is the potential energy of the particle and E is the total energy. Now what I am seeing is that this exact same relation is being used in quantum...
In QM, states evolve in time by action of the Time Evolution Unitary Operator, U(t,t0). Without the action of this operator, states do not move forward in time. Yet even stationary states, like an eigenstate of energy, still contain a time variable – they oscillate in time at a fixed...
Homework Statement
A particle of mass m in one dimension has a potential:
$$V(x) =
\begin{cases}
V_0 & x > 0 \\
0 & x \leq 0
\end{cases}
$$
Find ##\psi(x)## for energies ##0 < E < V_0##, with parameters
$$k^2 = \frac{2mE}{\hbar^2}$$
and
$$\kappa^2 = \frac{2m(V_0 - E)}{\hbar^2}$$...
Hello Forum,
Just checking my correct understanding of the following fundamental concepts:
Stationary states: these are states represented by wavefunctions ##\Psi(x,y,z,t)## whose probability density function ##|\Psi(x,y,z,t)|^2 = |\Psi(x,y,z)|^2##, that is, the pdf is only a function of space...
My textbook in elementary Q.M. stated that orbital electrons in an atom must have stationary state
wavefunctions. Was this just a simplification, the truth being maybe that their wavefunctions can be
nonstationary for a little while, but soon decay into stationary ones? I’ve seen an answer...
TL;DR:
My professor asked me to graph the probability that a particle would be excited from the ground state to a stationary state with a certain energy E (y-axis) verse the energy of that new state (x-axis). I need help finding this probability as a function of E.
Probability=|<ΨE|P|Ψg>|2
P is...
Hello everyone,
I am wondering if the eigenstates of Hermitian operators, which represent possible wavefunctions representing the system, are always stationary wavefunctions, i.e. the deriving probability distribution function is always time invariant. I would think so since these eigenstates...
Hello everybody,
- In quantum mechanics, the state ## | \psi \rangle ## of a system that is in thermodynamic equilibrium can be expressed as a linear combination of its stationary states ## | \phi _n \rangle ## : $$ | \psi \rangle = \sum_n c_n | \phi _n \rangle $$
It permit us to express the...
Homework Statement
I'm going back through some homework as revision, and came across this problem. It was marked as correct, but now I'm thinking it's unconvincing...
For a particle in an infinite square well, with ##V = 0 , 0 \leq x \leq L##, prove that the stationary eigenstates are...
I was watching some Steve Spicklemire QM videos and had a question/check my knowledge..
When we measure a the state of a system, say a particle in a box or a quantum harmonic oscillator (QSHO), we "collapse" the superposition of the system and end up with one eigenstate and one eigenvalue...
two questions:
1. besides using Ehrenfests theorem, is there another way of showing that the expectation value of momentum is zero in a stationary state ? (I don't see it when simply applying the definition on the stationary solution)
2. If we have a state that is a superposition of...
Hello there,
I am just starting quantum physics with the textbook by griffiths. My lecturer has told me that the set of functions representing stationary states in Hilbert space forms an orthogonal set. He was however unable to prove it. Furthermore he said that it is not always the case, but...
Tell me if the following is correct. For a simple infinite square well potential, the solutions to the Schrodinger equation are \Psi_n(x)=\sqrt{\frac{2}{a}}sin(\frac{n\pi x}{a}), then you plug in the appropriate value for n and operate on the function accordingly to get your observables.
Then...
Homework Statement
let [1> and [2> mutually orthogonal states (eigenstates of some Hermitian operator).
the Hamiltonian operator is given by H=c[1><2]+c[2><1], where c is a real number.
(a) calculate the eigenstates and corresponding eigenvalues of H
(b) if the initial state of the system...
According to Griffiths QM book, after he derived the stationary state solutions to the Schrodinger equation for a particle in an infinite potential well, which are just functions of sine, he claims that these stationary solutions are orthogonal and complete.
I agree that they are orthogonal...
Homework Statement
Prove the following theorum:
If V(x) is an even function (that is, ##V(-x) = V(x)##) then ## \psi (x) ## can always be taken to be either even or odd.
Hint: If ## \psi ## satisfies equation [1.0] for a given E, so too does ## \psi (-x) ## and hence also the odd and...
Homework Statement
A particle of mass m in the infinite square well of width a at time t = 0 is in a linear superposition of the ground- and the first excited- eigenstates, specifically it has the wave function
$$| \Psi(x,t) > = A[ | \psi_1 > + e^{i \phi} | \psi_2 >$$
Find the...
I always thought they were the same, but now I am reading a question that says "which of he following time-independent functions describe stationary states of the corresponding quantum systems?"
Is there something I am missing? It's written like there is something to solve, but to me it seems...
So in Griffith's (ed. 2 page 37) there's an equation that says that
**pretend the h's are h-bars...I don't know Latex very well**
ih\frac{1}{\varphi}\frac{d\varphi}{dt}=-\frac{h^{2}}{2m}\frac{1}{\psi}\frac{d^{2}\psi}{dx^{2}}+V
Since in this simplified case V where is a function of x alone...
Homework Statement
dJ(x,t)/dx = -dY^2 / dt , where y is the wave equation, and the d's represent partial derivatives. I want to make an assumption that I can describe the wave equation as a stationary state, so my question is the following:
What is the definition of a stationary state...
I'm pretty confused by the rules regarding the total energy, the kinetic energy, and potential of a QM system.
Does the total energy have to be positive or greater than zero? And if so, why not? I don't really understand what it means to have a negative total energy of a system I guess. I...
What I know: In stationary states the time dependence is factored out so it is of the form phi(q) * e^(-i omega t), thus in its appearance there is no wave function spread. However I recall from texts that wave packet spread is considered a universal phenomena in quantum mechanics, so I am...
Hi, when we consider an N-particle (assume non-interacting) system, say putting them in a box; why do we always say the states of the system (e.g. when counting them to find the "density of states as function of energy") are just the products of single particle stationary states (i.e. energy...
Hi I know that stationary states in a system with an even potential energy function have to be either even or odd.
Why does the ground state have to be even, and not odd? This is asserted in Griffiths, page 298.
Really confused by this one.
Homework Statement
I'm given that a tritium atom, with one proton and two neutrons in the nucleus, decays by beta emission to a helium isotope with two protons and one neutron in the nucleus. During the decay, the atom changes from hydrogen to singly-ionized...
The problem is to obtain the stationary states for a free particle in three dimensions by separating the variables in Schrödinger's equation.
So take
\psi(\mathbf{r},t) = \psi_1(x) \psi_2(y) \psi_3(z) \phi(t)
and substitute it into the time-dependent Schrödinger equation. For the...
I am trying to understand the nature of uncertainty relations in quantum mechanics. I am looking specifically at a relation between energy and position uncertainty... the book that I am reading hints that this relationship has no meaning in a stationary state. Why would that be?
First, read this; it's a two slit experiment carried out recently that is supposed to weaken the Copenhagen interpretation.
http://en.wikipedia.org/wiki/Double-slit_experiment#Shahriar_Afshar.27s_experiment
What are the stationary states physically? The problem with the stationary states...