At its core, the Hamiltonian of the harmonic oscillator involves the operator that is the product of the raising and lowering operators, ˆ a† ˆ

a. We had shown that the physically sensible eigenvalues of this operator are just non-negative integers, n, and so this operator is also called the number operator as it counts the number of the energy eigenstate that is occupied. The number operator is often denoted as ˆN = ˆ a† ˆ a.

Further, up to non-trivial commutation relations, ˆ a† ˆ a is like the squared magnitude of the operator ˆ a and so this suggests another representation of the creation and annihilation operators in analogy with complex numbers. If we define a Hermitian phase operator Θˆ , then we can write

Note that the square root of the number operator is well-defined as all of its eigenvalues are non-negative. We’ll study properties of this representation of the creation and annihilation operators here, and further in Exercise 6.6.

Transcribed Image Text:

t = e-ie, = eie . (6.58)