Please answer 7, 8, and 9. The first page is provided for context. The Fock operator â is defined by(1)||2hwhere î and p are the position and momentum operators, respectively.1. Write dowm ât in terms of î and p.2. Show that(ât +â)2mw(2)p=i2ħmw(ât – â)(3)hold.3. Show that the cannonical communation relation, [â, p = iħ, yields the so-called bosoniccommutation relation,(â, ât] = 1.(4)4. Show that the Hamiltonian of the SHO,is written as22m(*+)Ĥ = hw Ñ +(5)where N = âtâ is called the number operator.5. Show that Ñ is Hermitian. Suggestion: Use the identity from Exercise #1, (QR)† =6. A normalized vector |0) (so that (0|0) = 1) is defined to satisfy â|0) = 0. With thisthe following vectors are constructed:|n) = A, (ât)" |0) for n == 0, 1,2, ...(6)where An are constant with A, = 1. Compute Nn) for n =are the eigenvectors of N, i.e., Ñ|N) is proportional to |N). Find the eigenvalues ofN from the proportionality.0, 1,2,3 to show that theseCS Scanned with CamScanner 7. Find A, by normalizing |n). Suggestion: Compute (n|n) for n = 0, 1, 2,3 then find thepattern. Use the identity (Qflg) = (SIQ†|9) for Q = ât.8. Show that |n) are the eigenvectors of H, i.e.,Ĥ \n) = En \n)(7)is satisfied. Find the energy eigenvalue En. Knowing that Ĥ is an observable, whatcan you tell about (m|n) for m + n?9. Using your results, show that the following formula holds:ât |n) = Vn+1 |n+1)â n) = Vn In – 1)(8a)(8b)Do this either by giving a general proof or by showing that they hold for n == 0,1,2,3.10. Let's go back to Eq.(2). What is the SI unit of the coefficient? Does it make2mwsense to you?11. Show that the square of position operator is(âtât + â†â + ââ† + ââ)2mw(9)12. Compute (0|£|0) and (0|â²|0). This can be done very efficiently if you use Eq.(8)13. Find the expression of the momentum operator square, p, in terms of the Fock oper-ators.Compute (Op|0) and (0|* |0).

7. Find A, by normalizing |n). Suggestion: Compute (n|n) for n = 0,1, 2,3 then find the pattern. Use the identity (Qflg) = (flQ*]g) for Q = ât. 8. Show that |n) are the eigenvectors of H, i.e., ÎĤ \n) = En \n) (7) is satisfied. Find the energy eigenvalue En. Knowing that H is an observable, what can you tell about (m/n) for m + n? 9. Using your results, show that the following formula holds: ât \n) = Vn +1 n+ 1) â n) = Vn n – 1) (8a) (8b) Do this either by giving a general proof or by showing that they hold for n = 0,1,2, 3.

7. Find A, by normalizing |n). Suggestion: Compute (n|n) for n = 0,1, 2,3 then find the pattern. Use the identity (Qflg) = (flQ*]g) for Q = ât. 8. Show that |n) are the eigenvectors of H, i.e., ÎĤ \n) = En \n) (7) is satisfied. Find the energy eigenvalue En. Knowing that H is an observable, what can you tell about (m/n) for m + n? 9. Using your results, show that the following formula holds: ât \n) = Vn +1 n+ 1) â n) = Vn n – 1) (8a) (8b) Do this either by giving a general proof or by showing that they hold for n = 0,1,2, 3.