PH3101: Quantum Mechanics IITutorial 1: Mathematical Framework of Quantum

Mechanics

Question 1: Two vectors in a three-dimensional complex vector space are definedby:

|A〉 =

2−7i

1

, |B〉 =

1 + 3i48

Let a = 6 + 5ia) Compute a |A〉, a |B〉, and a (|A〉+ |B〉). Show that a (|A〉+ |B〉) = a |A〉+

a |B〉.b) Find the inner products 〈A|B〉, and 〈B|A〉.

Question 2:(a) Find the Hermitian conjugate of

1. 〈φ|A|ψ〉〈ψ|

2. A|ψ〉〈φ| − iAB

(b) Consider the following matrix and vectors:

A =

2 i 0−i 1 10 1 0

, |χ〉 =1

3

1ii

, |λ〉 =1

3

i−i2

,

1. Is A Hermitian? Are the vectors normalized?

2. Verify whether the vector |χ〉 is an eigenstate of A. If yes, find the corre-sponding eigenvalue.

3. Evaluate 〈λ|A|χ〉.

Question 3: Let’s consider three observables in quantum mechanics with 2 × 2matrix representations as follow:

σx =

(0 11 0

), σy =

(0 −ii 0

), σz =

(1 00 −1

)a) Find the normalized eigenvectors of these observables and the corresponding

eigenvalues. Is there any degeneracy?

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b) The matrix representations given above have eigenvectors of σz as the basis.By using eigenvectors of σx as a new basis, give the matrix representations of theobservables σx, σy, and σz. Can you give some comments about the result?

Question 4: Consider three orthonormal kets |φ1〉, |φ2〉, and |φ3〉.

1. Find the normalization factor of the state |Ψ〉 = |φ1〉+ 2 |φ2〉 − i√

5 |φ3〉.

2. Find the value of b such that the state |Ψ〉 is orthogonal to the state |X〉 =2 |φ1〉+ |φ2〉+ b |φ3〉.

3. Find the expectation value of an operator A2 with respect to |Ψ〉 and |X〉,where A |φn〉 = (2n − 1) |φn〉.

Question 5: Show that the set of all square-integrable functions is a vector space.[Hint: You may want to use the Cauchy-Schwartz inequality to show that the sumof two square-integrable functions is itself square-integrable.] Is the set of allnormalized functions a vector space?

Question 6: Prove that if 〈u,v〉1 and 〈u,v〉2 are two inner products on a vectorspace V, then the quantity 〈u,v〉 = 〈u,v〉1 + 〈u,v〉2 is also an inner product onV.

Question 7: Prove that if 〈u,v〉 is the Euclidean inner product on Rn and if Ais an n× n matrix, then

〈u, Av〉 = 〈ATu,v〉

[Hint: Use the fact that 〈u,v〉 = u · v = vTu]

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