2 1 900 X k 6 0 k \u02c6 x3 2k 1900 Xk 60 3 2 \u03b4k3 3 2 2 \u03b4 k1 2 k 1900 32 2 3 3 2 2

# 2 1 900 x k 6 0 k ˆ x3 2k 1900 xk 60 3 2 δk3 3 2 2

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2)=-1900Xk6=0kˆx302k=-1900Xk6=032δk,3+322δk,12k=-19003223+32221=-190014+98=-117200-1.5278×10-34. Unlike the variational principle, perturbation theory can be applied straightforwardly to excitedstates.Use it to approximate, to 2ndorder, the anharmonic oscillator’s first excited state’senergy.Starting with the action of ˆx3on|1i:ˆx3|1i=12224|4i+ 62|2i+ 3|0i+ 0|0i=3|4i+ 3|2i+322|0ikˆx31=3hk|4i+ 3hk|2i+322hk|0i=3δk,4+ 3δk,2+322δk,0The first-order correction to the energy of the anharmonic oscillator first excited state isE(1)1=-1301 ˆx31=-130h1|3|4i+ 3|2i+322|0i=-1303h1|4i+ 3h1|2i+322h1|0i=0
The second-order correction to the energy of the anharmonic oscillator first excited state isE(2)1=Xk6=1k-130ˆx312(1 +12)-(k+12)=-1900Xk6=1kˆx312k-1=-1900Xk6=13δk,4+ 3δk,2+322δk,02k-1=-1900324-1+|3|22-1+32220-1=-19001 + 9-98=-717200Therefore our approximation of the energy of the first excited state of the anharmonic oscillatorisE1E(0)1+E(1)1+E(2)1=32+ 0-717200=1072972001.49014