PLEASE ANSWER #14. The first page is provided for context. 7. Find An by normalizing |n). Suggestion: Compute (n|n) for n=0,1,2,3 then find thepattern. Use the identity (Qflg) = (fIQ*lg) for Q = ât.8. Show that n) are the eigenvectors of H, i.e.,ÊĤ
) = En
)(7)is satisfied. Find the energy eigenvalue En. Knowing that H is an observable, whatcan you tell about (m|n) for m + n?9. Using your results, show that the following formula holds:ât
) = Vn+1 |n+ 1)â
) = Vn |n – 1)(8a)(8b)Do this either by giving a general proof or by showing that they hold forn== 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 + âtâ + âât + ââ)2mw(9)12. Compute (0|î|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.14. Compute (0|p|0) and (0|p²|0).CS Scanned with CamScanner The Fock operator â is defined by(±+ mu)å -â =(1)2hwhere î and p are the position and momentum operators, respectively.1. Write down ât in terms of £ and p.2. Show that(ât + â)2mw(2)||himwp = i(ât – â)2(3)%3Dhold.3. Show that the cannonical communation relation, [ê, f] = iħ, yields the so-called bosoniccommutation relation,(â, ât] = 1.(4)4. Show that the Hamiltonian of the SHO, H =2mis written asÎĤ = f (N +)(5)where N = âtâ is called the number operator.5. Show that Ñ is Hermitian. Suggestion: Use the identity from Exercise #1, (QR)t =6. A normalized vector |0) (so that (0|0) = 1) is defined to satisfy â|0) = 0. With thisthe following vectors are constructed:|n) = An (ât)" |0) for n =0,1,2,...(6)where A, are constant with Ao = 1. Compute N|n) 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

Question

PLEASE ANSWER #14.  The first page is provided for context. 7. Find An by normalizing |n). Suggestion: Compute (n|n) for n=0,1,2,3 then find thepattern. Use the identity (Qflg) = (fIQ*lg) for Q = ât.8. Show that n) are the eigenvectors of H, i.e.,ÊĤ 
) = En 
)(7)is satisfied. Find the energy eigenvalue En. Knowing that H is an observable, whatcan you tell about (m|n) for m + n?9. Using your results, show that the following formula holds:ât 
) = Vn+1 |n+ 1)â 
) = Vn |n – 1)(8a)(8b)Do this either by giving a general proof or by showing that they hold forn== 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 + âtâ + âât + ââ)2mw(9)12. Compute (0|î|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.14. Compute (0|p|0) and (0|p²|0).CS Scanned with CamScanner The Fock operator â is defined by(±+ mu)å -â =(1)2hwhere î and p are the position and momentum operators, respectively.1. Write down ât in terms of £ and p.2. Show that(ât + â)2mw(2)||himwp = i(ât – â)2(3)%3Dhold.3. Show that the cannonical communation relation, [ê, f] = iħ, yields the so-called bosoniccommutation relation,(â, ât] = 1.(4)4. Show that the Hamiltonian of the SHO, H =2mis written asÎĤ = f (N +)(5)where N = âtâ is called the number operator.5. Show that Ñ is Hermitian. Suggestion: Use the identity from Exercise #1, (QR)t =6. A normalized vector |0) (so that (0|0) = 1) is defined to satisfy â|0) = 0. With thisthe following vectors are constructed:|n) = An (ât)&#34; |0) for n =0,1,2,...(6)where A, are constant with Ao = 1. Compute N|n) 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

Accepted Answer

The operators can be expressed as,

14. Compute (0|p|0) and (0\f*|0).