The above constraints show that a product of two Hermitian operators is Hermitian only if they mutually commute. The operator AB − BA is called the commutator of A and B and is denoted by [A, B]. If A and B commute, then [A, B] = 0.

For A and B hermitian operators, show that AB is hermitian if and only if A and B commute. but for hermitian operators, the RHS is BA which is equal to the LHS only when 0 = AB − BA = [A,B].

In classical mechanics you can measure any two observables simultaneously. In quan- tum mechanics, only variables whose (Hermitian) operators commute can be observed simultaneously.

The symmetrization of the classical physical quantity is necessary to en- sure that the resulting operator A is Hermitian. Moreover, neither XP nor PX are Hermitian, since (XP)† = PX.

The kinetic energy operator is given by: So, we have: You can use equation to check for the hermiticity of the Hamiltonian by just replacing with . Once you do this, you will find that the condition in the equality is satisfied and therefore the Hamiltonian is indeed Hermitian.

The expectation values of Hermitian operators are always real. The eigenvectors of Hermitian operators span the Hilbert space. The eigenvectors of Hermitian operators belonging to distinct eigenvalues are orthogonal.

Since we have shown that the Hamiltonian operator is hermitian, we have the important result that all its energy eigenvalues must be real. In fact the operators of all physically measurable quantities are hermitian, and therefore have real eigenvalues.

The adjoint of an operator A may also be called the Hermitian conjugate, Hermitian or Hermitian transpose (after Charles Hermite) of A and is denoted by A∗ or A† (the latter especially when used in conjunction with the bra–ket notation).

Hermitian operators play an integral role in quantum mechanics due to two of their proper- ties. First, their eigenvalues are always real. This is important because their eigenvalues correspond to phys- ical properties of a system, which cannot be imaginary or complex.

Theorem: The Hermitian conjugate of the product of two matrices is the product of their conjugates taken in reverse order, i.e. ]ij = [RHS]ij .

Here we know that A=iddx is Hermitian, saying A has its adjoint A∗=iddx.

Why is the derivative (d/dx) thought of as a linear operator instead of a function of functions? if we take the derivative of some function f(x) (d/dx(f(x))), then we get a new function f'(x). However d/dx is considered to be a linear operator. …

21 0. a) Consider the operator x d/dx(where 1st d/dx acts on the function, then x acts on the resulting function by simply multiplying by x )acting on the set of functions of a real variable x for x>0.

Like the word ladder suggests, these operators move eigenvalues up or down. They are used in angular momentum to rise or lower quantum numbers and quantum harmonic oscillators to move between energy levels.