Commuting Hermitian Matrices
From the above, the set of powers of an Hermitian matrix all commute with each
other, and have a common set of eigenvectors (but not the same eigenvalues,
obviously). In fact it is not difficult to show that any two Hermitian
The Position Operator and Its Eigenstates
The position is just the co-ordinate x, manifestly always real, and a Hermitian
operator.
Proof:
We shall make clear that in this context we regard x as an operator by writing it with a
little hat, It is equally c
First consider the nondegenerate case: A has all eigenvalues distinct. The eigenkets
of A, appropriately normalized, form an orthonormal basis in the space.
Write
Now
so
Note also that, obviously, V is unitary:
We have established, then, that for a Hermit
Differential Operators: the Momentum Operator on L2
Our task now is to recast this old approach of differential operators acting on wave
functions in the equivalent Dirac language. Lets begin with the simplest, the
momentum operator. First, we need to sho
Functions on the Infinite Line
What happens if we take the analysis of the previous section and let L go to
infinity? This is parallel to the analysis (two lectures back) of going from Fourier
series to the Fourier transform, the sum over a series of plan
Infinite Dimensions
The observation that the set of solutions to Schrdingers equation satisfies some of
the basic requirements of a vector space, in that linear combinations of solutions give
another solution to the equation. Furthermore, Schrdingers equa
Diagonalizing a Hermitian Matrix
As discussed above, a Hermitian matrix is diagonal in the orthonormal basis of its set
of eigenvectors:
, since
If we are given the matrix elements of A in some other orthonormal basis, to
diagonalize it we need to rotate
Eigenvectors of a Hermitian Matrix Span the Space
Well now move on to the general case: what if some of the eigenvalues of A are the
same? In this case, any linear combination of them is also an eigenvector with the
same eigenvalue. Assuming they form a b
Diagonal Determinants
Notice that the diagonal term in the determinant
the leading two orders in the polynomial
lower order terms too). Equating the coefficient of
in
generates
, (and some
here with that
,
Putting
in both the determinantal and the polynom
Electron in a Box
As a preliminary to discussing functions on the infinite line, it is worth considering
those restricted to the finite interval (0, L) and vanishing at the two ends. These are
precisely the conditions satisfied by the electron-in-a-box wa
The Basic Rules of Quantum Mechanics
Any quantum mechanical wave function must be normalizable, because the norm
represents the total probability of finding the particle (or, more generally, the system)
somewhere in its phase space, so
First Basic Rule: a