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PC3130/3201 QUANTUM MECHANICS TUTORIAL 4 March 2000 1. (a) Prove that the trace of an operator A, TrA = u i i A ui , is independent of the particular othonormal basis {ui >} that is chosen for its evaluation. 2. (b) Prove that Tru > < v= < v u >. (a) Show that for = 1, the matrices representing 1 and 3 are respectively 0 1 0 1 1 0 1 2 0 1 0 1 0 0 3 0 0 0 0 0 1 (b) Show that if an operator commutes with any two components of the angular momentum operator , it must commute with the remaining ~ components. (c) 3. Show that Tr i = 0, i = 1, 2, 3. Let a and a+ be two Hermitian conjugate operators such that [a, a+] = 1. We put N = a+ a. Show that [N, aj ] = -j aj; [N, a+j ] = j a+j (j integer > 0) 4. In question (3), if { n > } are the normalized eigenkets of the operator N, i.e. N n > = n n > n = 0, 1, 2, ..... < n n > = nn Show that (i) < i x2 j > 2m j 1 j 2ij 2 j j 1ij 2 ( 2 j 1)ij (no sum over j) where i, j are non-negative integers. (ii) (iii) 1 m ( 2 j 1)ij 2 (no sum over j) i p 2 j ( x ) 2 x 2 x 2 ( p ) 2 j 1 j 2ij 2 j j 1ij 2 ( 2 j 1) 2m 1 m ( 2 j 1) 2 1 ( 2 j 1) E j / 2 Here, i > and j > are eigenstates of N with eigenvalues i and j respectively and Thus, xp 1/ 2 x (a a ) , 2m (iv) p ( m / 2)1/ 2 i( a a ) Show that the operator N can be expressed as N= n n n n 5. Suppose operators a and a satisfy the following relations aa a a 1 , a 2 (a ) 2 0 These are known as fermionic operators in contrast with the bosonic operators discussed in the lectures. (i) Can the operator a be Hermitian ? (ii) Prove that the only possible eigenvalues for the operator N a a are zero and one. Explain. 6. Consider the Hamiltonian for the two dimensional motion of a particle of mass in a harmonic oscillator potential: H (i) (ii) 1 1 p12 p 22 + 2 ( x12 x 22 ) 2 2 1 Show that the energy eigenvalues are given by E n (n ) , where 2 n n1 n2 , with n1 , n2 0,1, 2,......... . Express the operator 3 x1 p 2 x2 p1 in terms of the annihilation operators a1 1 2 X 1 iP1 , a2 1 2 X 2 iP2 and the corresponding creation operates a 1 and a 2 . Here X i2 Pi 2 1 p i2 , i = 1, 2. Show that H, 3 0 . xi2 ,