\varepsilon is only a unit of energy: the constants in front of D_2, collected in one symbol
Potential on the diagonal; kinetic energy links neighbors through -\varepsilon (the Hückel picture, too)
Example: a box with three grid points
Three interior points, \psi = 0 at the walls. Find the energies E = \lambda\,\varepsilon of \;H = \varepsilon\begin{pmatrix} 2 & -1 & 0 \\ -1 & 2 & -1 \\ 0 & -1 & 2 \end{pmatrix}.
Solve \det(H/\varepsilon - \lambda I) = 0 along the first row: each entry times the 2\times2 determinant left after crossing out its row and column, signs +\,-\,+:
Sample the exact ground state \sin(\pi x/L) at the three points: \mathbf{v} = \big(\sin\tfrac{\pi}{4},\ \sin\tfrac{\pi}{2},\ \sin\tfrac{3\pi}{4}\big) \propto (1,\ \sqrt{2},\ 1). Is it an eigenvector of H?
Yes, with \lambda = 2 - \sqrt{2} (the middle entry 2\sqrt{2} - 2 is (2-\sqrt{2})\sqrt{2}). With \Delta x = L/4, \varepsilon = 8\hbar^2/mL^2 and E_1 = 4.69\,\hbar^2/mL^2, close to the exact 4.93.
More points, better answers
Eigenvectors are the exact states sampled on the grid; the levels converge from below
Why conjugate? Take \lvert\psi\rangle = \begin{pmatrix} 1 \\ i \end{pmatrix} and find \langle\psi\vert\psi\rangle.
\langle\psi\rvert = (1,\ -i), so \langle\psi\vert\psi\rangle = 1\cdot 1 + (-i)(i) = 2: a real, positive length squared. Without the conjugate, 1 + i^2 = 0: a nonzero vector of length zero. So a constant leaves a bra conjugated: \langle c\,\phi \vert \psi \rangle = c^*\langle \phi \vert \psi \rangle.
Sandwiches: matrix elements
An operator acts on a ket: \hat{A}\lvert\psi\rangle is the matrix times the column
A bra in front turns it into a number, the matrix element:
For A = \begin{pmatrix} 2 & 1 \\ 1 & 2 \end{pmatrix}, \mathbf{e}_1 = (1, 0) and \mathbf{e}_2 = (0, 1), compute \langle \mathbf{e}_1 \vert A \vert \mathbf{e}_2 \rangle.
A\,\mathbf{e}_2 = (1,\ 2), then \mathbf{e}_1 \cdot (1,\ 2) = 1 = A_{12}: sandwiching unit vectors picks out one entry of the matrix.
Count the shapes
Bra = row, operator = square, ket = column
Inner sizes must match; the outer sizes give the result: a number, or a new ket
Diagonalizing a Hamiltonian
The entries of a matrix depend on the axes; the operator does not. In perpendicular unit axes \mathbf{u}_1, \mathbf{u}_2, \dots each entry is a sandwich, A'_{mn} = \langle \mathbf{u}_m \vert A \vert \mathbf{u}_n \rangle
In the axes of its eigenvectors a matrix is diagonal: eigenvalues on the diagonal, zeros elsewhere
DiagonalizingH means finding these axes; the diagonal lists the energies, {2 - \sqrt{2}},\ 2,\ {2 + \sqrt{2}} for the three-point box
E, U = np.linalg.eigh(H): the axes are the columns of U, and U.T @ H @ U is diagonal
Dirac notation: one language, three dialects
integral
Dirac
matrices (numpy)
inner product
\int\phi^*\psi\,dx
\langle\phi\vert\psi\rangle
np.vdot(phi, psi) * dx
operator acts
\hat{A}\psi(x)
\hat{A}\lvert\psi\rangle
A @ psi
matrix element
\int\phi^*\hat{A}\psi\,dx
\langle\phi\vert\hat{A}\vert\psi\rangle
np.vdot(phi, A @ psi) * dx
eigenproblem
\hat{A}\phi_n = a_n\phi_n
\hat{A}\lvert n\rangle = a_n\lvert n\rangle
a, phi = np.linalg.eigh(A)
new axes
\int\phi_m^*\hat{A}\phi_n\,dx
\langle \phi_m \vert \hat{A} \vert \phi_n \rangle
U.conj().T @ A @ U * dx
The same objects in three dialects: integrals for functions, brackets on paper, matrices on a computer
Two matrix properties to watch
Mirror: each matrix equals its own conjugate transpose. Next lecture: Hermitian, which makes eigenvalues real
Order: for matrices AB \neq BA in general. Next lecture: commutators, and why x and p are never sharp together
Takeaway
Sample a function and it becomes a vector; every operator becomes a matrix. Position is diagonal, derivatives link neighbors, and the Hamiltonian is a matrix whose eigenvalues are the allowed energies. Dirac notation writes integrals and matrices in one language, and in the axes of its own eigenvectors a matrix is diagonal. Next: which matrices can be observables, and what happens when two of them do not commute.
