Probability of finding a system an eigenstate

[krobben92]

Homework Statement

As the homework problem is written exactly:
Consider the quantum mechanical system with only two stationary states |1> and |2> and energies E₀ and 3E₀, respectively. At t=0, the system is in the ground state and a constant perturbation <1|V|2>=<2|V|1>=E₀ is switched on. Calculate the probability of finding the system in the state |2>.

Homework Equations

H=V (I'm just assuming these states must be such that there is no kinetic energy and E=V, but that's just my guess - professor lacks communication skills at times).

|c₁|² + |c₂|² = 1
P(|2>)=|c₂|² (thus, I need c₂ to solve the problem!)

The Attempt at a Solution

Well, I'm completely confused as to why he gave us <1|V|2>=<2|V|1>=E₀, what that even means, and where I'm supposed to retrieve the probability coefficients which is all I need to solve the problem.

 

[nrqed]

Homework Statement

As the homework problem is written exactly:
Consider the quantum mechanical system with only two stationary states |1> and |2> and energies E₀ and 3E₀, respectively. At t=0, the system is in the ground state and a constant perturbation <1|V|2>=<2|V|1>=E₀ is switched on. Calculate the probability of finding the system in the state |2>.

Homework Equations

H=V (I'm just assuming these states must be such that there is no kinetic energy and E=V, but that's just my guess - professor lacks communication skills at times).

|c₁|² + |c₂|² = 1
P(|2>)=|c₂|² (thus, I need c₂ to solve the problem!)

The Attempt at a Solution

Well, I'm completely confused as to why he gave us <1|V|2>=<2|V|1>=E₀, what that even means, and where I'm supposed to retrieve the probability coefficients which is all I need to solve the problem.

Hi there!

Have you learned non degenerate perturbation theory? This is the formalism required to do this problem. What formula have you seen for perturbation theory?

 

[krobben92]

No, at least not that I know of. I'm familiar with chapters 1-4 in Griffiths, if that helps.

 

[vanhees71]

Also you are given the complete information about the unperturbed Hamiltonian!

 

[krobben92]

Also you are given the complete information about the unperturbed Hamiltonian!

Right, I get that I have all of the matrix elements of the Hamiltonian, but I'm not sure how that's going to help me determine that probability of finding the system in state |2>. Also, we've never covered non degenerate perturbation theory, so is there another more elementary way to solve this problem?

 

[vela]

Homework Statement

As the homework problem is written exactly:
Consider the quantum mechanical system with only two stationary states |1> and |2> and energies E₀ and 3E₀, respectively. At t=0, the system is in the ground state and a constant perturbation <1|V|2>=<2|V|1>=E₀ is switched on. Calculate the probability of finding the system in the state |2>.

Homework Equations

H=V (I'm just assuming these states must be such that there is no kinetic energy and E=V, but that's just my guess - professor lacks communication skills at times).

No, this isn't right. You add V to the existing Hamiltonian, so H=H₀+V. What are the matrix representations of H₀, V, and H? You should be able to figure out the rest from there.

|c₁|² + |c₂|² = 1
P(|2>)=|c₂|² (thus, I need c₂ to solve the problem!)

The Attempt at a Solution

Well, I'm completely confused as to why he gave us <1|V|2>=<2|V|1>=E₀, what that even means, and where I'm supposed to retrieve the probability coefficients which is all I need to solve the problem.