Download Quantum Mechanics I - Homework II Practice | PHYS 943 and more Assignments Quantum Mechanics in PDF only on Docsity! Physics 943 - Homework 2 Department of Physics Due February 13, 2009 University of New Hampshire 1 Comprehensive Practice The spin-1/2 particle once more. A spin-1/2 particle is in the state χ given by: χ = A ( 2 1− i ) . (1) The Pauli matrices are defined by: σx = ( 0 1 1 0 ) , σy = ( 0 −i i 0 ) , σz = ( 1 0 0 −1 ) . (2) (a) Find the value of A. (b) Find the eigenvalues and eigenvectors for the y-component of spin, Sy. (c) For a particle in the state χ given in Eq. (1), find the values of Sy that can be measured and the probabilities for measuring each of them. What is the average value of Sy in the state χ? (d) Find the root mean square value of Sy (∆Sy). 2 Warm-up, more with ijk. A tensor field, commonly in physics referred to as a “tensor”, is an extension of scalar, vector, matrix. A tensor with an array of n indices (Tijk....) is said to be of rank-n. The Levi-Civita tensor, or totally asymmetric tensor, is a rank 3 tensor defined by: ijk = 1, when j = 1, j = 2, k = 3 and all cyclic permutations of these −1, when j = 2, j = 1, k = 3 and all cyclic permutations of these 0, all other combinations (3) 1 A useful lemma about tensors: If Aij is anti-symmetric (Aij = −Aji) and Sij is symmetric ( Sij = Sji) then∑ ij AijSij = 0. (a) Show that ∑ k ijklmk = δilδjm − δimδjl. (b) Show that ∑ jk ijkljk = 2δil. (c) Show that ∑ ijk ijkijk = 6 (d) Write an expression for ( Â× B̂ )2 in terms of a sum over components and simplify this expression using previous results. Here  and B̂ are operators with three components, i.e. Â1, Â2, Â3. (e) Use the previous results to proof the identity: L̂2 = (x̂× p̂)2 = x̂2p̂2 − (x̂ · p̂)2 + ih̄ (x̂ · p̂) (4) Note that [x̂, p̂] = ih̄. Be careful in this part not to make sign errors. Hint: If you find the wrong sign for one of the terms, perhaps you forgot that ∑ k = 3? 3 Silver atoms, once again. This problem is about the polarization states of a silver atom. As discussed in class, silver atoms have only two independent polarizations, so in this problem we are dealing with a two-dimensional Hilbert space and its bases (|Z+〉 , |Z−〉), etc. Stern-Gerlach experiments with multiple magnetic gaps tell us that | 〈Z ± |X±〉 |2 = | 〈Z ± |Y±〉 |2 = | 〈X ± |Y±〉 |2 = 1 2 , (5) where |Z+〉 is a normalized ket vector representing a quantum state in which the z-component of the atom’s magnetic moment has definite value µz = +µ0 ≡ +eh̄/2Mec. |Z−〉 represents the state that has µz = −µ0 and same for the |X±〉 and |Y±〉. 2
