# Do expectation values vary with time?

• Darkmisc
In summary: If A is not conserved, then <A(t)> is time-dependent.In summary, the conversation discusses the concepts of probability conservation and expectation values. It is stated that probability conservation is expressed by the partial derivative of probability density with respect to time being equal to zero. The conversation also mentions that expectation values can depend on the position of a particle and can vary with time depending on the probability distribution and the dynamics of the system. It is noted that if an operator is conserved, then its expectation value will remain constant over time.
Darkmisc
I'm a bit confused about the nature of probability conservation and expectation values.

According to probability conservation,

$\frac{∂P(r,t)}{∂t}$=0.

Does that mean that expectation values e.g. <x>, <p> and <E> depend only on the position of the particle and not on time?

Thanks

Im not sure what your $P$ means, but the conservation law is actually expressed
$$\frac{\partial \rho}{\partial t}=-\frac{\partial j_i}{\partial x_i}$$
With $\rho$ the probability density, and $j$ the probability current density. They are defined
$$\rho=\Psi^{*}\Psi \quad j_i=\Psi^{*}\partial_i \Psi-\Psi\partial_i \Psi^{*}$$
More on this

P is the normalised probability of finding the particle somewhere in a given volume.

Do expectation values vary with time with ρ and j as defined?

Total probability is obviously conserved, but expectation values depend on the probability distribution, which can of course vary with time.

If $A$ is an operator on the Hilbert space of the physical states $|\psi \rangle$, then

$\frac{\textbf{d}}{\textbf{d}t}\langle \psi| A |\psi\rangle = -\frac{i}{\hslash} \langle\psi|[A,H] |\psi\rangle\, .$

From this follow that the expectation value of an operator over an Hamiltonian eigenstate is constant as it should.
Moreover the mean energy is always conserved.

Ilm

Darkmisc said:
Do expectation values vary with time with ρ and j as defined?
They can, but this depends in the wave function.

Think about a free Gaussian wave packet centered at x=0 for t=0; the expectation value is of course <x> = 0.

Now you can "boost" this wave such that it moves along the x-axis. Its width grows with time, but of course the normalization P(-∞,+∞) i.e. the probability to find the particle somewhere in the interval (-∞,+∞) remains P = 1 = const. Due to the boost the wave packet moves along the x-axis, i.e. the expectation value for x becomes time-dependent: <x(t)> = p/m * t. The expectation value for the momentum <p(t)> = p = const. b/c there is no potential (we started with a free wave packet).

That means that the the answer to you questioin depends on:
- state wave function ψ you are looking at
- the dynamics, i.e. the Hamiltonian H of your system
- the observable A for which you want to calculate <A(t)>

If A is conserved, i.e. [H,A]=0 the <A(t)> = const.

## 1. Do expectation values change over time?

Yes, expectation values can change over time. This is because expectation values are dependent on the state of the system, and the state of a system can change over time due to various factors such as external influences or internal dynamics.

## 2. How do expectation values vary with time in quantum mechanics?

In quantum mechanics, expectation values are described by the time-dependent Schrodinger equation. This equation allows us to calculate the time evolution of a system's wave function, which in turn determines the expectation values of observables at different points in time. Therefore, expectation values can vary with time in a predictable manner in quantum mechanics.

## 3. Can expectation values decrease with time?

Yes, expectation values can decrease with time. This can occur if the system is undergoing a decay process or if it is being affected by external forces that cause a decrease in the observable quantity being measured.

## 4. Do expectation values always approach a steady state over time?

Not necessarily. In some cases, expectation values may approach a steady state over time, but in other cases, they may exhibit periodic or chaotic behavior. This depends on the specific dynamics of the system being studied.

## 5. Can the uncertainty in expectation values increase with time?

Yes, the uncertainty in expectation values can increase with time. This is because as the system evolves, its state can become more uncertain, leading to a wider range of possible expectation values for a given observable. This is described by the Heisenberg uncertainty principle in quantum mechanics.

Replies
1
Views
2K
Replies
6
Views
4K
Replies
3
Views
1K
Replies
1
Views
913
Replies
2
Views
678
Replies
12
Views
2K
Replies
15
Views
950
Replies
20
Views
3K
Replies
1
Views
1K
Replies
13
Views
2K