QUANTUM OPERATORS
an arbitrary complete set of orthonormal base states|φi〉and using equation
(19.11), is as follows:
〈ψ|B^2 |ψ〉=〈ψ|B× 1 ×B|ψ〉=∑i〈ψ|B|φi〉〈φi|B|ψ〉=∑i〈ψ|B|φi〉(
〈φi|B|ψ〉∗)∗=∑i〈ψ|B|φi〉(
〈ψ|B†|φi〉)∗=∑i〈ψ|B|φi〉〈ψ|B|φi〉∗, sinceBis Hermitian,=∑i|〈ψ|B|φi〉|^2 ≥ 0.We note, for future reference, that the HamiltonianHfor the s.h.o. is the sum of
two terms each of this form and therefore conclude that〈ψ|H|ψ〉≥0 for all|ψ〉.
The energy spectrum of the simple harmonic oscillatorLet the normalised ket vector|n〉(or|En〉) denote thenth energy state of the s.h.o.
with energyEn. Then it must be an eigenstate of the (Hermitian) HamiltonianH
and satisfy
H|n〉=En|n〉with〈m|n〉=δmn.Now consider the stateA|n〉and the effect ofHupon it:
HA|n〉=AH|n〉−ωA|n〉, using (19.42),=AEn|n〉−ωA|n〉=(En−ω)A|n〉.ThusA|n〉is an eigenstate ofHcorresponding to energyEn−ωand must be
some multiple of the normalised ket vector|En−ω〉,i.e.
A|En〉≡A|n〉=cn|En−ω〉,wherecnis not necessarily of unit modulus. Clearly,Ais an operator that
generates a new state that is lower in energy byω; it can thus be compared to
the operatorD, which has a similar effect in the context of thez-component of
angular momentum. Because it possesses the property of reducing the energy of
the state byω, which, as we will see, is one quantum of excitation energy for the
oscillator, the operatorAis called anannihilation operator. Repeated application
ofA,mtimes say, will produce a state whose energy ismωlower than that of
the original:
Am|En〉=cncn− 1 ···cn−m+1|En−mω〉. (19.44)