Action of a and a+ on 1n

Action of 𝑎 ^ a ^ & 𝑎 ^ † a ^ † on ∣ 𝑛 ⟩ ∣n⟩ : 𝐻 ∣ 𝑛 ⟩ = 𝐸 𝑛 ∣ 𝑛 ⟩ H∣n⟩=E n ​ ∣n⟩ [ 𝑎 ^ , 𝑎 ^ † ] = − ℏ 𝜔 [ a ^ , a ^ † ]=−ℏω ⇒ 𝑎 ^ 𝐻 = 𝑎 ^ 𝐻 − 𝑎 ^ ℏ 𝜔 ⇒ a ^ H= a ^ H− a ^ ℏω 𝑎 ^ 𝐻 ∣ 𝑛 ⟩ = ( 𝑎 ^ 𝐻 − 𝑎 ^ ℏ 𝜔 ) ∣ 𝑛 ⟩ a ^ H∣n⟩=( a ^ H− a ^ ℏω)∣n⟩ = 𝑎 ^ 𝐻 ∣ 𝑛 ⟩ − 𝑎 ^ ℏ 𝜔 ∣ 𝑛 ⟩ = a ^ H∣n⟩− a ^ ℏω∣n⟩ = 𝑎 ^ 𝐸 𝑛 ∣ 𝑛 ⟩ − 𝑎 ^ ℏ 𝜔 ∣ 𝑛 ⟩ = a ^ E n ​ ∣n⟩− a ^ ℏω∣n⟩ = ( 𝐸 𝑛 − ℏ 𝜔 ) 𝑎 ^ ∣ 𝑛 ⟩ =(E n ​ −ℏω) a ^ ∣n⟩ Therefore, 𝑎 ^ ∣ 𝑛 ⟩ = ∣ 𝑛 − 1 ⟩ a ^ ∣n⟩=∣n−1⟩ 𝐻 ∣ 𝑛 − 1 ⟩ = ( 𝐸 𝑛 − ℏ 𝜔 ) ∣ 𝑛 − 1 ⟩ H∣n−1⟩=(E n ​ −ℏω)∣n−1⟩ ⇒ 𝐸 𝑛 − 1 = 𝐸 𝑛 − ℏ 𝜔 ⇒E n−1 ​ =E n ​ −ℏω Action of 𝑎 ^ † a ^ † 𝑎 ^ † ∣ 𝑛 ⟩ = ∣ 𝑛 + 1 ⟩ a ^ † ∣n⟩=∣n+1⟩ 𝐻 ∣ 𝑛 + 1 ⟩ = ( 𝐸 𝑛 + ℏ 𝜔 ) ∣ 𝑛 + 1 ⟩ H∣n+1⟩=(E n ​ +ℏω)∣n+1⟩ Similar for 𝑎 ^ † a ^ † [ 𝑎 ^ , 𝑎 ^ † ] = 𝑎 ^ + ℏ 𝜔 [ a ^ , a ^ † ]= a ^ +ℏω 𝑎 ^ 𝐻 = 𝑎 ^ 𝐻 + 𝑎 ^ ℏ 𝜔 a ^ H= a ^ H+ a ^ ℏω 𝐻 ∣ 𝑛 + 1 ⟩ = ( 𝐸 𝑛 + ℏ 𝜔 ) ∣ 𝑛 + 1 ⟩ H∣n+1⟩=(E n ​ +ℏω)∣n+1⟩ ⇒ 𝐸 𝑛 + 1 = 𝐸 𝑛 + ℏ 𝜔 ⇒E n+1 ​ =E n ​ +ℏω ⇒ 𝐸 𝑚 − 1 = ( 𝑚 − 1 ) ℏ 𝜔 ⇒E m−1 ​ =(m−1)ℏω 𝑎 ^ † 𝐻 = 𝑎 ^ † 𝐻 + 𝑎 ^ † ℏ 𝜔 a ^ † H= a ^ † H+ a ^ † ℏω 𝐻 = 𝑎 ^ † 𝐻 + 𝑎 ^ † ℏ 𝜔 H= a ^ † H+ a ^ † ℏω 𝐸 𝑚 − 1 = ( 𝑚 − 1 ) ℏ 𝜔 E m−1 ​ =(m−1)ℏω a ^ ∣n⟩ is the eigenstate of 𝐻 ^ H ^ with eigenvalue ( 𝐸 𝑛 + ℏ 𝜔 ) (E n ​ +ℏω) which is corresponding to a state raised by unity w.r.t. ∣ 𝑛 ⟩ ∣n⟩: ⇒ 𝑎 ^ ∣ 𝑛 ⟩ = 𝐶 ∣ 𝑛 − 1 ⟩ ⇒ a ^ ∣n⟩=C∣n−1⟩ ⇒ 𝑎 ^ † ∣ 𝑛 ⟩ = 𝐶 + ∣ 𝑛 + 1 ⟩ ⇒ a ^ † ∣n⟩=C + ​ ∣n+1⟩ Let 𝐶 C and 𝐶 + C + ​ be notation wise. These are action equations. ⇒ 𝑎 ^ ∣ 𝑛 ⟩ = 𝐶 ∣ 𝑛 − 1 ⟩ ⇒ a ^ ∣n⟩=C∣n−1⟩ ⇒ ⟨ 𝑛 ∣ 𝑎 ^ † = ⟨ 𝑛 − 1 ∣ 𝐶 ∗ ⇒⟨n∣ a ^ † =⟨n−1∣C ∗ ⇒ ⟨ 𝑛 ∣ 𝑎 ^ † 𝑎 ^ ∣ 𝑛 ⟩ = ⟨ 𝑛 − 1 ∣ 𝑒 𝑖 𝜙 𝐶 ∗ 𝐶 ∣ 𝑛 − 1 ⟩ ⇒⟨n∣ a ^ † a ^ ∣n⟩=⟨n−1∣e iϕ C ∗ C∣n−1⟩ ⇒ ⟨ 𝑛 ∣ 𝑛 ⟩ = ∣ 𝐶 ∣ 2 ⟨ 𝑛 − 1 ∣ 𝑛 − 1 ⟩ ⇒⟨n∣n⟩=∣C∣ 2 ⟨n−1∣n−1⟩ ⇒ 𝑛 ⟨ 𝑛 ∣ 𝑛 ⟩ = ∣ 𝐶 ∣ 2 ⟨ 𝑛 − 1 ∣ 𝑛 − 1 ⟩ ⇒n⟨n∣n⟩=∣C∣ 2 ⟨n−1∣n−1⟩ ⇒ 1 ⇒ ⟨ 𝑛 ∣ 𝑛 ⟩ = ⟨ 𝑛 − 1 ∣ 𝑛 − 1 ⟩ = 1 𝑛 ⟨ 0 ∣ 0 ⟩ = 𝛿 𝑚 𝑛 ⇒1⇒⟨n∣n⟩=⟨n−1∣n−1⟩= n 1 ​ ⟨0∣0⟩=δ mn ​ ⇒ 𝑛 = ∣ 𝐶 ∣ 2 ⇒n=∣C∣ 2 C − ​ =± m ​ ∣ 𝐶 − 1 ∣ = 𝑚 ∣C−1∣= m ​ 𝐶 − C − ​ is norm so it must be + 𝑣 𝑒 +ve and real ⇒ 𝑎 ^ ∣ 𝑛 ⟩ = 𝑚 ∣ 𝑛 − 1 ⟩ ⇒ a ^ ∣n⟩= m ​ ∣n−1⟩ ⇒ 𝑎 ^ † ∣ 𝑛 ⟩ = 𝐶 + ∣ 𝑛 + 1 ⟩ ⇒ a ^ † ∣n⟩=C + ​ ∣n+1⟩ ⟨ 𝑛 ∣ 𝑎 ^ † = ⟨ 𝑛 + 1 ∣ 𝐶 + ∗ ⟨n∣ a ^ † =⟨n+1∣C + ∗ ​ ⟨ 𝑛 ∣ 𝑎 ^ 𝑎 ^ † ∣ 𝑛 ⟩ = ⟨ 𝑛 + 1 ∣ ∣ 𝐶 + ∣ 2 ∣ 𝑛 + 1 ⟩ ⟨n∣ a ^ a ^ † ∣n⟩=⟨n+1∣∣C + ​ ∣ 2 ∣n+1⟩ ⟨ 𝑛 ∣ ( 𝑎 ^ + 𝑎 ^ + 1 ) ∣ 𝑛 ⟩ = ∣ 𝐶 + ∣ 2 ⟨ 𝑛 + 1 ∣ ∣ 𝑛 + 1 ⟩ ⟨n∣( a ^ + a ^ +1)∣n⟩=∣C + ​ ∣ 2 ⟨n+1∣∣n+1⟩ ⟨ 𝑛 ∣ ( 𝑛 + 1 ) ∣ 𝑛 ⟩ = ∣ 𝐶 + ∣ 2 ⟨ 𝑛 + 1 ∣ ∣ 𝑛 + 1 ⟩ ⟨n∣(n+1)∣n⟩=∣C + ​ ∣ 2 ⟨n+1∣∣n+1⟩ ∣ 𝐶 + ∣ 2 = 𝑛 + 1 ⇒ ∣C + ​ ∣ 2 =n+1⇒ E n ​ =(n+ 2 1 ​ )ℏω 𝑛 = 0 , 1 , 2 , 3 , … n=0,1,2,3,… ⇒ 1 2 ℏ 𝜔 , 3 2 ℏ 𝜔 , 5 2 ℏ 𝜔 , 7 2 ℏ 𝜔 ⇒ 2 1 ​ ℏω, 2 3 ​ ℏω, 2 5 ​ ℏω, 2 7 ​ ℏω ∴ 𝑎 ^ ∣ 𝑛 ⟩ = 𝑚 ∣ 𝑛 − 1 ⟩ ∴ a ^ ∣n⟩= m ​ ∣n−1⟩ 𝑎 ^ 2 ∣ 𝑛 ⟩ = 𝑎 ^ 𝑚 ∣ 𝑛 − 1 ⟩ a ^ 2 ∣n⟩= a ^ m ​ ∣n−1⟩ 𝑎 ^ 2 ∣ 𝑛 ⟩ = 𝑚 𝑎 ^ ∣ 𝑛 − 1 ⟩ = 𝑚 𝑚 − 1 ∣ 𝑛 − 2 ⟩ a ^ 2 ∣n⟩= m ​ a ^ ∣n−1⟩= m ​ m−1 ​ ∣n−2⟩ 𝐸 𝑛 + 1 − 𝐸 𝑛 = ℏ 𝜔 E n+1 ​ −E n ​ =ℏω For example: 𝑎 ^ ∣ 0 ⟩ = 0 a ^ ∣0⟩=0 𝑎 ^ 𝜓 0 ( 𝑥 ) = 0 a ^ ψ 0 ​ (x)=0 ( 𝑚 𝜔 2 ℏ 𝑥 + 𝑖 𝑝 ^ 2 ℏ 𝑚 𝜔 ) 𝜓 0 ( 𝑥 ) = 0 ( 2ℏ mω ​ ​ x+ 2ℏmω ​ i p ^ ​ ​ )ψ 0 ​ (x)=0 𝑚 𝜔 ℏ 𝑥 𝜓 0 ( 𝑥 ) + 𝑖 ( − 𝑖 ℏ 𝑑 𝑑 𝑥 𝜓 0 ( 𝑥 ) ) / 2 ℏ 𝑚 𝜔 = 0 ℏ mω ​ xψ 0 ​ (x)+i(−iℏ dx d ​ ψ 0 ​ (x))/ 2ℏmω ​ =0 ⇒ ℏ 𝑑 𝜓 0 𝑑 𝑥 / 2 ℏ 𝑚 𝜔 = − 𝑚 𝜔 ℏ 𝑥 𝜓 0 ⇒ℏ dx dψ 0 ​ ​ / 2ℏmω ​ =− ℏ mω ​ xψ 0 ​ ∫ 𝑑 𝜓 0 𝜓 0 = − 𝑚 𝜔 ℏ ∫ 𝑥 𝑑 𝑥 + Const. ∫ ψ 0 ​ dψ 0 ​ ​ =− ℏ mω ​ ∫xdx+Const.