Time evolution of a quantum system

In summary, time evolution is a crucial concept in quantum mechanics that describes how the state of a system changes over time and allows for predictions of its future state. It differs from classical mechanics in that it is probabilistic and can exhibit phenomena such as superposition and entanglement. The time evolution of a quantum system is influenced by factors such as the initial state, the Hamiltonian, and interactions with other systems or the environment. Unlike classical mechanics, the time evolution of a quantum system cannot be reversed by simply reversing the direction of time. Observation and measurement techniques, such as spectroscopy and time-resolved measurements, can be used to study the time evolution of a quantum system.
  • #1
CAF123
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Homework Statement


Let the time evolution of a system be determined by the following Hamiltonian: $$\hat{H} = \gamma B \hat{L}_y$$ and let the system at t=0 be described by the wave function ##\psi(x,y,z) = D \exp(-r/a)x,## where ##r## is the distance from the origin in spherical polars. Find the state of the system at any time t. Deduce the expectation value of ##\hat{L}_z## in the state ##\psi##.

Homework Equations


In Cartesians, $$\hat{L}_y = -i\hbar \left(z \frac{\partial}{\partial x} - x \frac{\partial}{\partial z}\right)$$

The Attempt at a Solution


Computing the Hamiltonian gives the new wave function ##-Di \hbar \gamma B z \exp(-r/a)##. To obtain the time dependence, by solving the spatial-independent Schrodinger equation, $$\psi(x,y,z,t) = \psi(x,y,z)\exp(-Et/\hbar).$$ I am not sure if this is correct since ##\psi(x,y,z)## is not an eigenstate of ##\hat{L}_y## and hence not of ##\hat{H}## either.
Thanks.
 
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  • #2
You're right. You can't use that expression for the time evolution of a generic state. It only works for eigenstates.

You need to express the initial wave function as a linear combination of the eigenstates of the Hamiltonian because you know how to write down the time evolution of an eigenstate. So the first thing you need to do is figure out what those eigenstates are.
 
  • #3
vela said:
You need to express the initial wave function as a linear combination of the eigenstates of the Hamiltonian because you know how to write down the time evolution of an eigenstate. So the first thing you need to do is figure out what those eigenstates are.

I forgot to mention in a previous part of the question, we are given eigenfunctions of ##\hat{L}_z## and ##\hat{L}_y##. For ##\hat{L}_y## they are ##\phi (x,y,z) = f(r)(z\pm ix)##, corresponding to eigenvalues ##\pm \hbar##. So these are also the eigenfunctions of ##\hat{H}## with e.values ##\pm \gamma B \hbar##. Hence, $$\psi (x,y,z) = \sum_{n=1}^2 c_n \phi (x,y,z) = c_1 f(r) (z+ix) + c_2 f(r) (z-ix)$$ and the ##c_i## are obtained by projecting the wave function ##\psi## onto this basis: $$c_n = \langle \psi | \phi\rangle = \int_{\text{all space}} x \exp(-r/a)f(r)(z+ix)\,d\tau$$ and similarly for the other coefficient. Am I supposed to do this integral?
 
  • #4
That would be one way, but there's a simpler way. Remember the goal is to express ##\psi## in terms of the eigenfunctions. You want to choose ##c_1## and ##c_2## such that
$$c_1 f(r) (z+ix) + c_2 f(r) (z-ix) = D e^{-r/a}x$$ holds for all x, y, and z. You should be able to see what f(r) is by inspection. Collect terms on the lefthand side and then match coefficients for x and z between the two sides to get a system of equations for ##c_1## and ##c_2##.
 
  • #5
vela said:
That would be one way, but there's a simpler way. Remember the goal is to express ##\psi## in terms of the eigenfunctions. You want to choose ##c_1## and ##c_2## such that
$$c_1 f(r) (z+ix) + c_2 f(r) (z-ix) = D e^{-r/a}x$$ holds for all x, y, and z. You should be able to see what f(r) is by inspection. Collect terms on the lefthand side and then match coefficients for x and z between the two sides to get a system of equations for ##c_1## and ##c_2##.
I see, so in the end, $$\psi (x,y,z,t) = \frac{D}{2} f(r) (z+ix) \exp(-iE_1t/\hbar) - \frac{D}{2} f(r) (z-ix)\exp(-iE_2t/\hbar)$$ where ##E_1## and ##E_2## are the energy eigenvalues corresponding to the each function in the basis.

To compute the expectation value of ##\hat{L}_z## in the state ##\psi## does that mean deal with the wave function at time t=0?, so compute ##\langle \hat{L}_z \rangle = \int \psi^* \hat{L}_z \psi\,d\tau = \int D^2 \exp(-r/a)x \hat{L}_z \exp(-r/a)x\,d\tau\,?##

Edit: I computed this integral, and since f(r)(z+/-ix) are not e.functions of ##\hat{L}_z##, I get terms that vanish and the integral is zero.
 
Last edited:
  • #6
The problem may be asking you to calculate the expectation value of ##\hat{L}_z## as a function of time. It's not clear from the wording, though, so you might want to ask your professor.
 

1. What is the significance of time evolution in quantum systems?

The concept of time evolution in quantum systems is crucial for understanding the behavior and properties of these systems. It describes how the state of a quantum system changes over time, and allows us to predict the future state of the system based on its current state. Time evolution is also important for studying the dynamics of quantum systems, which can provide insights into the underlying physical processes and principles governing these systems.

2. How does time evolution differ from classical mechanics?

In classical mechanics, time evolution is deterministic and follows Newton's laws of motion. However, in quantum mechanics, time evolution is described by the Schrödinger equation, which is a probabilistic equation. This means that the state of a quantum system at a given time cannot be precisely determined, but only the probability of it being in a certain state can be calculated. Additionally, quantum systems can exhibit phenomena such as superposition and entanglement, which have no classical analog.

3. What factors influence the time evolution of a quantum system?

The time evolution of a quantum system is influenced by several factors, including the initial state of the system, the Hamiltonian (energy operator) of the system, and any interactions with other systems or the environment. The Hamiltonian plays a crucial role in determining the energy levels and dynamics of a quantum system, while interactions can cause entanglement and affect the system's evolution over time.

4. Can the time evolution of a quantum system be reversed?

In classical mechanics, the time evolution of a system can be reversed by reversing the direction of time. However, in quantum mechanics, this is not the case. The laws of quantum mechanics are time-symmetric, meaning that the equations governing the evolution of a system are the same whether time is moving forward or backward. This means that the time evolution of a quantum system cannot be reversed by simply reversing the direction of time.

5. How is the time evolution of a quantum system observed or measured?

The time evolution of a quantum system can be observed or measured through various techniques, such as spectroscopy, interferometry, and time-resolved measurements. These techniques involve interacting with the system and measuring its state at different points in time. The results of these measurements can then be used to reconstruct the time evolution of the system and gain insights into its properties and behavior.

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