PX262 - M3 - dirac notation

it is a very economical and powerful notation
associated with each wavefunction, $ψ (x)$ , is a state vector, $| ψ ⟩$ , called a 'ket'
with each complex conjugate, there is an associated 'bra', $⟨ ψ |$

\begin{matrix} \int ϕ^{*} (x^{'}) ψ (^{'} x) d x^{'} = ⟨ ϕ | ψ ⟩ \\ (⟨ ϕ | ψ ⟩)^{*} = ⟨ ψ | ϕ ⟩ \end{matrix}

for an expression involving an operator, eg: an expectation value:

\int ψ^{*} (x^{'}) \hat{A} ψ (x^{'}) d x^{'} = ⟨ ψ | A | ψ ⟩

the eigenvalue equation is written as:

\hat{A} | a ⟩ = λ_{a} | a ⟩

where, state $| a ⟩$ is labelled by its eigenvalue

orthonormality:

⟨ a_{1} | a_{2} ⟩ = δ_{a_{1}, a_{2}}

the expansion process:

\begin{matrix} | ψ ⟩ = \sum_{a} c_{a} | a ⟩ \\ c_{a_{1}} = ⟨ a_{1} | ψ ⟩ \\ | ψ ⟩ = \sum_{a} | a ⟩ ⟨ a | ψ ⟩ \\ \sum_{a} | a ⟩ ⟨ a | = 1 \end{matrix}

using this, a lot of quantum mechanical problems can be recast in the language of matrices
this can be used to make approximations and models
the schrödinger equation:

\hat{H} | ψ ⟩ = i ℏ \frac{d}{d t} | ψ ⟩

where, the representation has not yet been specified

choosing some basis of states, $| a ⟩ :$

| ψ ⟩ = \sum_{a} | a ⟩ ⟨ a | ψ ⟩

\begin{matrix} \sum_{a} \hat{H} | a ⟩ ⟨ a | ψ ⟩ = i ℏ \sum_{a} | a ⟩ \frac{d}{d t} ⟨ a | ψ ⟩ \\ \sum ⟨ a_{1} | \hat{H} | a ⟩ ⟨ a | ψ ⟩ = i ℏ \sum_{a} ⟨ a_{1} | a ⟩ \frac{d}{d t} ⟨ a | ψ ⟩ \end{matrix}

ie. multiplied the schrodinger equation on the left by $⟨ a_{1} |$

\sum_{a} H_{a, a} c_{a} (t) = i ℏ \frac{d}{d t} c_{a_{1}} (t)

(\begin{matrix} H_{11} & H_{12} & \dots \\ H_{21} & ⋱ \\ ⋮ & ⋱ \end{matrix}) (\begin{matrix} c_{1} (t) \\ c_{2} (t) \\ ⋮ \end{matrix}) = i ℏ \frac{d}{d t} (\begin{matrix} c_{1} \\ c_{2} \\ ⋮ \end{matrix})

$H_{a, a}$ 's have the form of a square matrix
to illustrate, considering the case of two base states forming a complete set, eg: spin

| ψ ⟩ = c_{1} | ↑ ⟩ + c_{2} | ↓ ⟩

eg:

\begin{matrix} ⟨ ↑ | \hat{H} | ↑ ⟩ = E_{0} \\ H_{12} = ⟨ ↑ | \hat{H} | ↓ ⟩ = A = H_{21} \\ H_{22} = ⟨ ↓ | \hat{H} | ↓ ⟩ = E_{0} \\ (\begin{matrix} E_{0} & A \\ A & E_{0} \end{matrix}) (\begin{matrix} c_{1} \\ c_{2} \end{matrix}) = i ℏ \frac{d}{d t} (\begin{matrix} c_{1} \\ c_{2} \end{matrix}) \\ E_{0} c_{1} + A c_{2} (t) = i ℏ \frac{d c_{1}}{d t} \\ A c_{1} + E_{0} c_{2} (t) = i ℏ \frac{d c_{2}}{d t} \end{matrix}

it is useful to have a representation that diagonalizes the $H$ -matrix
to do this, the eigenstates and eigenvalues of the hamiltonian need to be found

\begin{matrix} \hat{H} | n ⟩ = E_{n} | n ⟩ \\ | ψ ⟩ = \sum_{n} c_{n} (t) | n ⟩ \\ and, c_{n} (t) = ⟨ n_{1} | ψ ⟩ \\ | ψ ⟩ = \sum_{n} | n ⟩ ⟨ n | ψ ⟩ \\ \sum_{n} \hat{H} | n ⟩ ⟨ n | ψ ⟩ = i ℏ \sum | n ⟩ \frac{d}{d t} ⟨ n | ψ ⟩ \\ \sum_{n} \hat{H} | n ⟩ c_{n} (t) = i ℏ \sum_{n} | n ⟩ \frac{d c_{n}}{d t} \end{matrix}

forming a matrix, $⟨ n_{1} | :$

\begin{matrix} \sum_{n} ⟨ n_{1} | \hat{H} | n ⟩ c_{n} (t) = i ℏ \sum_{n} ⟨ n_{1} | n ⟩ \frac{d c_{n}}{d t} \\ \hat{H} | n ⟩ = E_{n} | n ⟩ \\ ⟨ n_{1} | \hat{H} | n ⟩ = E_{n} ⟨ n_{1} | n ⟩ = E_{n} δ_{n n_{1}} \\ E_{n_{1}} c_{n_{1}} = i ℏ \frac{d c_{n_{1}}}{d t} \\ c_{n_{1}} (t) = c_{n_{1}} (0) \exp (- \frac{i E_{n_{1}} t}{ℏ}) \\ | ψ (t) ⟩ = \sum_{n} c_{n} (0) | n ⟩ \exp (- \frac{i E_{n_{1}} t}{ℏ}) \end{matrix}

with the initial $| ψ (0) ⟩$ state known, $| ψ (t) ⟩$ can be found