148 CHAPTER9. THEHARMONICOSCILLATOR
Proceedinginthisway,wefindaninfinitesetofeigenstatesoftheform
φn = cn(a†)nφ 0 n= 0 , 1 , 2 , 3 ,...En = ̄hω(n+1
2
) (9.50)
Now,howdoweknowthatthisisalltheeigenstatesthatthereare? Suppose,for
example, thatthere werean eigenstateφ′ withanenergy inbetween E 0 andE 1.
Ifthat werethecase,theapplyingtheloweringoperatortothestatewouldeither
annihilatethestate,orelsegiveaneigenstatewithanenergylowerthantheground
state. Sincethegroundstateexists,andisφ 0 bydefinition,itmeansthataφ′=0.
Buttheonlysolutiontothisequationisφ′=φ 0. Therefore,thereisnostatewith
energybetweenE 0 andE 1. Sosupposeinsteadtherewereastateφ′′withenergy
betweenEnandEn+1. Applyingtheloweringoperatortoφ′′lowerstheenergy by
integermultiplesof ̄hω,untilwereachastateφ′withanenergybetweenE 0 andE 1.
Butwehavejustseenthatthereisnosuchstate.Thismeansthatthereisnostate
withenergybetweenEnandEn+1. Sotheonlypossibleenergiesarethoseshownin
(9.50).
Still,howdoweknowthattheseenergiesarenon-degenerate?Suppose,e.g.,there
wasastateφ′ 1 +=φ 1 withenergyE 1. Applyingtheloweringoperator,wewouldget
asecondgroundstateφ′ 0 withenergyE 0. Butthen,sincethereisnoenergylower
thantheground-stateenergy,
aφ′ 0 = 0 =⇒φ′ 0 =φ 0 (9.51)whichimplies,since
φ′ 0 ∝aφ′ 1 (9.52)
thatφ′ 1 =φ 1 .ThisargumentiseasilygeneralizedtoanyenergyEn,andtheconclu-
sionisthatthereisonlyoneeigenstateforeachenergyeigenvalue.
Next,we need to find the normalization constants cn in(9.50). This isdone
iteratively. Beginfrom
c 0 = 1 (9.53)
andimposethenormalizationcondition
1 = <φn|φn>
= c^2 n<(a†)nφ 0 |(a†)nφ 0 >
= c^2 n<a†(a†)n−^1 φ 0 |(a†)nφ 0 >
= c^2 n<(a†)n−^1 φ 0 |a(a†)n|φ 0 > (9.54)Theideaistocommutetheaoperatortotheright,pastthea†operators, untilit
finallyoperateson,andannhilates,thegroundstateφ 0. Wehave
a(a†)n = (aa†)(a†)n−^1