Quantum Physics II (8.05) Fall 00 Assignment Readings The readings below will take you through the material for Problem Sets and 4. Cohen-Tannoudji Ch. II, III. Shankar Ch. 1 continues to be helpful. Sakurai 1.1-1.5 is in places a little bit more advanced than we need, but you will find much that is helpful. Griffiths: the rest of Ch.. The notes by R. L. Jaffe available on the web page. Problem Set 1. Application of the identity operator (6 points) Comment: The goal of this question is to learn how to handle the closure relation, or, equivalently, the identity operator. The results you obtain, for example that the trace of an operator does not depend on the choice of basis, should be familiar from linear algebra. Consider a Hilbert space spanned by an orthonormal basis { u 1, u, u,...}. Assume that the set of kets { v 1, v, v,...} is also a complete orthonormal basis. (a) Prove that u i  u i = v i  v i i where A is a linear operator. Use Dirac notation. Hint: insert the identity operator. (b) Define Tr( A) i u i A u A B i. Prove that Tr( B) = Tr( A) and that Tr( A B C) = Tr( C A B) = Tr( B C A). i 1
. Operators in a Three-State System (10 points) Suppose a quantum system has only three states of interest. This means that the Hilbert space is three dimensional. Suppose we choose a set of basis states that are eigenstates of an operator Â. Label the states 1,,, and suppose the eigenvalues of A are +a, 0, a respectively. Let B be another operator with the following properties: B 1 = b and B = ib. In this problem, a and b are both real and positive. (a) Write a matrix representation for  in the 1,, basis. (b) Suppose B = 0. What must B be if B is a Hermitian operator? Write a matrix representation for B in the 1,, basis. (c) What are the eigenvalues and normalized eigenstates of B? Write the eigenstates in the 1,, basis. (d) What are the matrix elements of A in the basis of eigenstates of B? [Please order the eigenstates of B in decreasing order by eigenvalue, as was done for A.]. Projectors (6 points) (a) Let  = ψ φ where ψ and φ are normalized vectors in a Hilbert space. Under what condition is A Hermitian? Under what condition is A a projector? Show that as long as ψ and φ are not orthogonal,  can be written as λp 1 P where P 1 and P are projection operators and λ is a constant. (b) Consider again the three state system of Problem, with basis states 1,,. The normalized state ψ is defined by 1 i 1 ψ = 1 + +. (1) Calculate the matrix representing (in the 1,, basis) the projection operator onto the state ψ. Verify that this matrix is Hermitian.
4. Unitary Operators (6 points) An operator U is called unitary if it has the property that where 1 is the identity operator. U U = U U = 1 (a) Show that if the norm of ψ is one, then the norm of U ψ is also one. Comment: This demonstrates that unitary transformations conserve probability. (b) Show that if  is hermitian, then exp(iâ) is unitary. Note: the operator exp A is defined by 1 1 exp  = 1 +  +  +  +...!! (c) Show that if the set { u i } is a complete orthonormal set, with u i u j = δ ij, then the set { v i }, where v i U u i, is also orthonormal. Comment: this means that a unitary operator acting on a set of orthonormal basis states yields another set of orthonormal basis states. 5. Unitary Transformations and Diagonalization. (8 points) Let A ij = u i A u j be matrix elements of a hermitian operator A in an orthonormal basis { u i }. Let v i be the eigenvectors of Â:  v i = a i v i. We choose { v i } to be an orthonormal set. (a) Let U be the unitary operator that rotates the u i basis into the v i basis: U u i = v i. Show that the matrix elements of U in the u i basis are U ij = u i v j. (b) Show that the matrix U = U ij diagonalizes the matrix A = A ij. That is, show that U AU is a diagonal matrix. (c) Show that the u i are eigenvectors of the rotated operator B = U A U. (d) Show that A and B have the same eigenvalues.
6. Operators in a two-dimensional Hilbert space (4 points) Consider a two-dimensional Hilbert space spanned by the orthonormal basis { +, }. In this basis, the matrix representation of the operators Ŝ x, Ŝ y and Ŝ z are given in terms of the Pauli matrices ( ) ( ) ( ) 0 1 0 i 1 0 σ x = ; σ y = ; σ i 0 z = 1 0 0 1 by h Ŝ i = σ i. (Throughout this problem, the indices i, j, k run over x, y, z.) (a) Write the operators Ŝ x, Ŝ y and Ŝ z in Dirac notation using +,, + and. (b) Are these operators Hermitian? (c) Consider the state 1 ψ = + +. Find ψ Ŝ z ψ, the expectation value for the measurement of the z-component of the spin on an ensemble of atoms each in the state ψ. Is the expectation value given by one of the eigenvalues of S z? [ Note: when it is understood what state is being discussed, one often writes ψ Ŝ z ψ as just Ŝ z.] (d) Since the state ψ is not an eigenstate of Ŝ z, we cannot predict the result of a measurement of the z-component of the spin with certainty. Having calculated the mean value of the probability distribution characterizing the result of this measurement, let us now calculate the uncertainty. Define ( ) ΔA Â A A = A. () Demonstrate the second equality in (). Show that if ψ were an eigenstate of A, then ΔA would be zero. Find ΔS z in the state ψ given in part (c). (e) Using the orthonormality of + and, prove that [ ] S i, S j = i h ε ijk S k, () k where ε xyz = 1, ε ijk changes sign if any two of its indices are exchanged, and ε ijk = 0 if any two of its indices are the same. [That is, ε xyz = ε yzx = ε zxy = 1; ε xzy = ε zyx = ε yxz = 1; and all other possibilities are zero.] 4
(f) Define the operator Ŝ Ŝx + Ŝ y + Ŝ z. [Note that this is an abuse of notation in the following sense. I have not (and will not) define an operator Ŝ. Ŝ should be read as the operator named S ; it cannot be read as (the operator named S).] Write S as a matrix. Show that [Ŝ, Ŝ i] = 0 for i = x, y, z. (g) Using the orthonormality of + and, prove that where { A, B} A B + BÂ. { } h S i, S j = δ ij (4) (h) Prove that an arbitrary Hermitian operator (in the two-dimensional Hilbert space considered in this problem) can be expressed as a linear combination of the identity operator and the three operators represented by the Pauli matrices σ i. (i) Find a unitary matrix U such that U σ y U = σ z. Find a different unitary matrix such that U σ z U = σ x. [Aside: From this and the result of Problem 5d, you can conclude that σ x,y,z all have the same eigenvalues, which is something you found explicitly on Problem Set.] [Second aside: the two unitary matrices you have constructed are examples of rotation matrices. You will see more of them on Problem Set 4.] (j) Throughout this problem, we have used the eigenvectors of σ z as our basis vectors, denoting them as + +; z and ; z. This is the conventional choice of basis. However, we could equally well have used the eigenvectors of σ y as our basis. What are the vectors that represent +; z and ; z in a basis with basis vectors +; y and ; y? 5