Obtain Time evolution from Hamiltonian

In summary, the conversation discusses a quantum system with a ##C^3## state space and a orthonormal base ##\{|1\rangle, |2\rangle, |3\rangle\}##, and the Hamiltonian operator that acts on it. The solution involves building the H matrix and obtaining the eigenvectors, and using a change of basis to express the vector ##|3\rangle## in terms of its eigenvectors. The final step is to apply the time-evolution to the eigenvectors to obtain the desired result. It is noted that the change of basis can be done using either ##S## or ##S^\dagger##.
  • #1
carllacan
274
3

Homework Statement


A quantum system with a ##C^3## state space and a orthonormal base ##\{|1\rangle, |2\rangle, |3\rangle\}## over which the Hamiltonian operator acts as follows:
##H|1\rangle = E_0|1\rangle+A|3\rangle##
##H|2\rangle = E_1|2\rangle##
##H|3\rangle = E_0|3\rangle+A|1\rangle##

Build H and obtain the eigenvalues and the eigenvectors.

If ##|\Phi((0)\rangle = |2\rangle## obtain ##|\Phi(t)\rangle##.

If ##|\Phi((0)\rangle = |3\rangle## obtain ##|\Phi(t)\rangle##.

Homework Equations

The Attempt at a Solution


I've managed to build the H matrix and obtain the eigenvectors.
The second part is ##|\Phi(t)\rangle = e^{-\frac{i}{\hbar}E_1t}|2\rangle##
It is the third part that I'm not sure. Should I build the change of basis matrix from the eigenvectors and apply it to the vector ##|3\rangle## expressed as ## |0 0 1\rangle##, so that I get its eigenvectors descomposition?
 
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  • #2
carllacan said:
It is the third part that I'm not sure. Should I build the change of basis matrix from the eigenvectors and apply it to the vector ##|3\rangle## expressed as ## |0 0 1\rangle##, so that I get its eigenvectors descomposition?
That should work, yes. Once you have the composition in eigenvectors, the time-evolution is easy to write down.
 
  • #3
Thanks mfb.

Just a little side question: I've done the change of basis with ##|3\rangle' = S|3\rangle S^{-1}##. Could I have done ##|3\rangle' = S^\dagger|3\rangle S##?
 
Last edited:

1. What is time evolution and why is it important?

Time evolution refers to the change of a physical system over time. It is important because it allows us to understand how a system behaves and evolves over time, which is crucial for predicting its behavior and making accurate scientific predictions.

2. What is a Hamiltonian and how is it related to time evolution?

The Hamiltonian is a mathematical operator that describes the total energy of a physical system. It is related to time evolution because it is used to determine the equations of motion for a system, which govern how the system evolves over time.

3. How is time evolution obtained from the Hamiltonian?

The time evolution of a system can be obtained by applying the time-dependent Schrödinger equation, which is derived from the Hamiltonian. This equation allows us to calculate the probability of finding a system in a particular state at a given time.

4. Can time evolution be calculated for all systems using the Hamiltonian?

In principle, yes. The Hamiltonian can be used to calculate the time evolution of any physical system, as long as the system is described by a wave function. However, for complex systems, the calculations may become extremely difficult and require advanced mathematical techniques.

5. Are there any limitations to using the Hamiltonian to obtain time evolution?

One limitation is that the Hamiltonian only works for systems that are described by a wave function. Additionally, it may not accurately describe certain quantum systems, such as those that involve strong interactions or high energies. In these cases, alternative approaches may be necessary.

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