Harmonic Oscillators

Abstract

Now we turn to the treatment of the oscillator using non-Hermitian operators, with an eye toward more general dynamical circumstances. For simplicity we use dimensionless variables q, p, and the non-Hermitian variables y, y† closely related to them, rather than dimensional x, p _x

$$\begin{array}{*{20}{c}} {x = \sqrt {{\frac{\hbar }{{M\omega }}}} q \to q = \frac{1}{{\sqrt {2} }}({{y}^{ + }} + y)} \\ {p(x)\sqrt {{\hbar M\omega }} p \to p = \frac{i}{{\sqrt {2} }}({{y}^{ + }} - y)} \\ {\frac{1}{{i\hbar }}[x,{{p}_{x}}] = \frac{1}{i}[q,p] = 1,\;[y,{{y}^{ + }}] = 1} \\ \end{array}$$

(7.1.1)

and express the energy in frequency units,

$$\begin{array}{*{20}{c}} {H = \frac{{p_{x}^{2}}}{{2M}} + \frac{1}{2}M{{\omega }^{2}}{{x}^{2}} = \hbar \omega (\frac{1}{2}{{p}^{2}} + \frac{1}{2}{{q}^{2}})} \\ { = \hbar \omega ({{y}^{ + }}y + \frac{1}{2}) \to \omega {{y}^{ + }}y,} \\ \end{array}$$

(7.1.2)

where, in addition, the irrelevant constant EquationSource MathType!MTEF!2!1!+- % feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca % aIXaaabaGaaGOmaaaacqWIpecAcqaHjpWDcuGH9aqpgaqcamaalaaa % baGaaGymaaqaaiaaikdaaaGaeqyYdChaaa!3ED9! \[\frac{1}{2}\hbar \omega \hat = \frac{1}{2}\omega \] $$ is subtracted, so that the eigenvalues of H are now ω, 2ω, 3ω, ....