Prolem Set1
1
Problem Set 1 on Quantum Mechanics II Preliminary Topics: Mathematical Formulation and Postulates 1. |φ n i are the eigenstates of a Hermitian operator H (H is, for example, the Hamiltonian of an arbitrary physical system). Assume that the states |φ n i form a discrete orthonormal basis. The operator U (m, n) is defined by: U (m, n)= |φ m ihφ n | (a) Calculate the hermitian adjoint of U (m, n). (b) Calculate the commutator [H, U (m, n)]. (c) Calculate Tr (U (m, n)). (d) Let A be an operator, with matrix with matrix representation in |φ n i bases. Prove the relation: A = X m,n A mn U (m, n). (e) Show that A pq = Tr AU † (p, q) . 2. Consider the Hamiltonian H of a particle in a one-dimensional problem defined by: H = 1 2m P 2 + V (X ); where X and P are bearing the ususal meaning satisfying the relation [X, P ]=˙ ı~. Now |φ n i’s are defined by H |φ n i = E n |φ n i , with discrete index n. (a) Show that hφ m | P |φ n i = α hφ m | X |φ n i ; where α is a coefficient which depends on the difference between E m and E n . Calculate α (hint: consider the commutator [X, H ] ). (b) From this, deduce, using the closure relation, the equation: X m (E n - E m ) 2 |hφ n | X |φ m i| 2 = ~ 2 M 2 hφ n | P 2 |φ n i . 3. A particle of mass m is in the ground state of the infinite square well. Suddenly the well expands to twice its original size,leaving the wave function (momentarily) undisturbed. The energy of the particle is now measured. (a) What is the most probable result? What is the probability of getting that result? (b) What is the next most probable result, and what is its probability? (c) What is the expectation value of the energy? 4. Using the relation hx|pi = (2π~) -1 exp (˙ ıpx), find the expressions hx| XP |ψi and hx| PX |ψi in terms of ψ(x). Can these results be found directly by using the fact that in the |xi representation, P acts like ~ ˙ ı d dx . 5. Find the momentum-space wave function Φ n (p, t) for the nth stationary state of the infinite square well. Construct |Φ n | 2 (it’s simplest to write separate formulas for odd and even n). Show that |Φ n | 2 is finite at p = ±nπ~/a.
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Transcript of Prolem Set1
Problem Set 1 on Quantum Mechanics II
Preliminary Topics: Mathematical Formulation and Postulates
1. |φn〉 are the eigenstates of a Hermitian operator H (H is, for example, the Hamiltonian of anarbitrary physical system). Assume that the states |φn〉 form a discrete orthonormal basis. Theoperator U(m,n) is defined by:
U(m,n) = |φm〉 〈φn|
(a) Calculate the hermitian adjoint of U(m,n).
(b) Calculate the commutator [H,U(m,n)].
(c) Calculate Tr (U(m,n)).
(d) Let A be an operator, with matrix with matrix representation in |φn〉 bases. Prove therelation:
A =∑m,n
AmnU(m,n).
(e) Show that Apq = Tr{AU †(p, q)
}.
2. Consider the Hamiltonian H of a particle in a one-dimensional problem defined by:
H =1
2mP 2 + V (X);
where X and P are bearing the ususal meaning satisfying the relation [X,P ] = ı̇~. Now |φn〉’sare defined by H |φn〉 = En |φn〉 , with discrete index n.
(a) Show that〈φm|P |φn〉 = α 〈φm|X |φn〉 ;
where α is a coefficient which depends on the difference between Em and En. Calculate α(hint: consider the commutator [X,H] ).
(b) From this, deduce, using the closure relation, the equation:
∑m
(En − Em)2 |〈φn|X |φm〉|2 =~2
M2〈φn|P 2 |φn〉 .
3. A particle of mass m is in the ground state of the infinite square well. Suddenly the well expandsto twice its original size,leaving the wave function (momentarily) undisturbed. The energy of theparticle is now measured.
(a) What is the most probable result? What is the probability of getting that result?
(b) What is the next most probable result, and what is its probability?
(c) What is the expectation value of the energy?
4. Using the relation 〈x|p〉 = (2π~)−1 exp (ı̇px), find the expressions 〈x|XP |ψ〉 and 〈x|PX |ψ〉 interms of ψ(x). Can these results be found directly by using the fact that in the |x〉 representation,P acts like ~
ı̇ddx .
5. Find the momentum-space wave function Φn(p, t) for the nth stationary state of the infinite squarewell. Construct |Φn|2 (it’s simplest to write separate formulas for odd and even n). Show that|Φn|2 is finite at p = ±nπ~/a.
