# Qantum mechanics condition for time independence

• Mechdude
In summary, the probability density for the given equation \psi(x,t) = \psi_1 e^{-i E_1 t/2} + \psi_2 e^{-iE_2 t/2} is |\Psi(x,t)|^2 = \psi(x,t)* \psi(x,t). The condition for the probability density to be time independent is if a dynamical variable is represented by a Hermitian operator A that commutes with H and contains no specific time dependence. In this case, the expectation value and variance of A will also be time independent, making it a constant of motion.
Mechdude

## Homework Statement

$$\psi(x,t) = \psi_1 e^{-i E_1 t/2} + \psi_2 e^{-iE_2 t/2}$$
under what conditions is the probability density time independent?

## Homework Equations

$$|\Psi(x,t)|^2 = \psi(x,t)* \psi(x,t)$$

## The Attempt at a Solution

i found a statement in pg 71 of prof Richard Fitzpatrick's notes on quantum mechanics (university of Texas at Austin) that says :
" If a dynamical variable is represented by some Hermitian operator A which
commutes with H (so that it has simultaneous eigenstates with H), and contains
no speciﬁc time dependence, then it is evident...that the expectation value and variance of A are time independent. In this sense,the dynamical variable in question is a constant of the motion."

is this the condition that is being sought for?
because by defenition getting the probability density will involve the relevant equation I've written down and the time variable exits out of the expression

Actually, what you have written in your attempt at a solution is referring to the expectation of some dynamical variable represented by the operator A.

All you have to do is plug in your equation for \Psi into the expression for the probability density. What do you find then and under what conditions do the time dependent part drop off?

## What is the condition for time independence in quantum mechanics?

The condition for time independence in quantum mechanics is that the Hamiltonian operator must commute with the time operator. This means that the Hamiltonian must not depend explicitly on time, and the system's energy must remain constant over time.

## How does time independence affect the behavior of quantum systems?

Time independence allows for the prediction and calculation of a system's behavior over time, as the system's energy remains constant and its state is determined solely by the Hamiltonian operator.

## What is the significance of the time-independent Schrödinger equation?

The time-independent Schrödinger equation is a fundamental equation in quantum mechanics that describes the behavior of a quantum system in a stationary state. It allows for the calculation of a system's wave function, which gives information about the system's energy and probability distribution over time.

## Can a system be time-independent and non-stationary?

No, if a system is time-independent, it must also be stationary. This means that the system's energy remains constant over time and its behavior can be predicted and calculated using the time-independent Schrödinger equation.

## How does the concept of time independence relate to the uncertainty principle?

The uncertainty principle states that it is impossible to know the exact position and momentum of a particle simultaneously. However, in a time-independent system, the Hamiltonian operator is conserved, meaning the energy is constant and therefore the momentum is also constant. This reduces the uncertainty in the momentum of a particle in a time-independent system.

Replies
10
Views
1K
Replies
2
Views
1K
Replies
2
Views
2K
Replies
1
Views
1K
Replies
9
Views
1K
Replies
3
Views
2K
Replies
1
Views
3K
Replies
1
Views
2K
Replies
2
Views
892
Replies
9
Views
1K