The Hamiltonian of a two-state system can be written H=( A1+B1ϵ B2ϵ B2ϵ A2 ) wher

The Hamiltonian of a two-state system can be written H=( ...

Key Concepts

Two-State Quantum Systems

This concept involves systems that can be described by two quantum states, forming the simplest non-trivial quantum mechanical systems. These systems are often modeled using a 2×2 Hamiltonian matrix, capturing both individual state energies and the interactions or transitions between the states.

Hamiltonian and Matrix Representation

The Hamiltonian represents the total energy of a quantum system and is typically expressed as a matrix when dealing with a finite basis of states. In a two-state system, a 2×2 Hamiltonian encapsulates the energy levels of each state (diagonal elements) and the coupling between them (off-diagonal elements). This matrix formulation is fundamental for analyzing eigenvalues and eigenstates of the system.

Exact Diagonalization

Diagonalizing a Hamiltonian matrix is an essential technique for finding the exact energy eigenvalues and eigenstates of a system. In the context of two-state systems, exact diagonalization provides closed-form solutions that fully describe the system's behavior, allowing for a precise understanding of its spectral properties.

Perturbation Theory

Perturbation theory is a method used to approximate the eigenvalues and eigenstates of a system when it is subject to a small external influence or parameter. It involves expanding the quantities of interest in a power series of a small parameter, making it possible to find corrections to the energies and states as the system deviates slightly from an exactly solvable model.

Degenerate vs Non-Degenerate Cases

In perturbation theory, the treatment of a system depends on whether the unperturbed eigenvalues are degenerate or not. In non-degenerate cases, simple perturbative corrections can be used since the energy levels are distinct, whereas degenerate cases require a more careful analysis (degenerate perturbation theory) to correctly account for potential mixing of states and avoid divergences in the correction terms.

Transcript

00:01 We're going to get the hamiltonian which is represented by the 1, 1 state minus 2, 2 plus 1, 2 plus 2, 1.

00:19 So this hamiltonian is going to have the matrix representation as a, 1, 1, 1, minus 1.

00:25 So essentially in order to find the eigenvalues and the eigencats corresponding to this hamiltonian, we're required to find the eigenvalues and eigencats for this very small matrix.

00:35 Remember that the eigenvalues for this matrix are going to be multiplied by the value for a and the eigencats, since we're going to normalize them at the very end, it really doesn't matter.

00:45 So, of course, this is assuming that the inner product of state 1 and 2 for i, j is going to be equal to delta i, j for any 1 and 2 combination that we can have.

01:05 Okay so over here in order to get the eigenvalues and eigencats for this matrix we know that the condition of the determinant of the matrix minus lambda times the identity needs to be equal to zero from here this becomes the determinant of 1 minus lambda 1 1 minus 1 minus lambda needs to be equal to zero 0, this is going to yield the characteristic equation, which is equal to lambda squared minus 2.

01:38 So the two possible eigenvalues that we're going to get are going to be lambda 1 and lambda 2 equal to square root of 2 and minus square root of 2.

01:50 So for lambda 1, the eigenket is going to be found from the condition that 1 minus lambda, 1, 1, minus 1, minus lambda multiplied by a vector b1 b2 which is in the base of the states 1 and 2 respectively means to be equal to the 0 0 we're going to call this eigen ket the plus eigen ket and we're going to replace the value for lambda 1 so this becomes 1 minus square root of 2 1 1 1 minus 1 minus square root of 2 times b1 and b2 equal to the 0, 0.

02:44 So from this set of equations, we know that b1 minus 1 minus square root of 2, in this case plus minus 1 minus square root of 2 b2, needs to be equal to 0.

03:00 So from here, we get that b1 is going to be equal to minus this result.

03:06 So 1 plus square root of 2 times b2.

03:18 What we're going to do is multiply all of this state by an n, which is going to be our normalization constant that we're going to find.

03:27 In this case, the requirement is that when we do plus state, plus state, this needs to be equal to 1.

03:37 And this will be equal to n squared times the inner products of each of the components in this case 1 plus square root of 2 squared of 1 1 plus 2 2 recall that we're going we're not going to have any cross turns because any contribution coming from 1 2 is going to be equal to 0 as as we stated at the very beginning of the problem.

04:11 So with this, since the initial states are guaranteed to have normalization, this becomes 1 and 1.

04:20 So 1 over n, sorry.

04:25 So this becomes 1 is equal to n squared times 1 plus the square root of 2 squared plus 1.

04:39 Thus, the value for the normalization for this plus state is going to be equal to 1 over 1 plus 1 plus the square root of 2 squared which we can reduce after doing some algebra into 1 over 2 times 2 plus the square root of 2 so with this our plus state is going to be equal to our normalization constant 1 over 2 times 2 times 2 plus the square root of 2 our vector over here that was 1 plus square root of 2 times state 1 plus state 2.

05:36 And the lambda plus associated to this value is going to be equal to a times square root of 2.

05:43 Now we can multiply the value for a because we know that the hamiltonian was our matrix multiplied by a.

05:52 Remember the eigenket is not going to get multiplied by the factor that we factorized out of the hamiltonian but the eigenvalue is going to be.

06:02 So now we're going to repeat the same procedure with the minus ket...

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