2. A state | describing the state of a two spin-1/2 systems is entangled when we can NOT write it...
3.16 For a spin-1/2 system undergoing Rabi oscillations, assume that the resonance condition holds. a) Solve the differential equations for the coefficients α + (r). Use your results to find the transformed state vector |ψ(t)) and the state vector ψ(t), assuming the most general b) Verify that a π-pulse (wit-T) produces a complete spin flip. Calculate both the c) Assume that the interaction time is such that ω/-: π/2. Find the effect on the system d) Discuss the differences between...
(L43*) Spin can be represented by matrices. Show that all three spin matrices l 0 2 0 -1 0),"2=2 1 have eigenvalues of +1/2h and -1/2h. Calculate the corresponding eigenfunctions which we will denote as α-and β-eigenfunctions corresponding to spin l/2 particles. Show that Sj can be determined by the commutation of the other two matrices sn and sm, n, maj. Prove that the (2×2) matrix sz-s' +ss+s, commutes with all spin matrices, ie. s2s,-sis-. Calculate the eigenvalues of s2....
1. We begin with a two state system with states labeled by |1) and [2). This may seem unphysical; however, there are many two state systems in quantum mechanics such spin 1/2 particles. The Hamiltonian we consider is (a) Compute the eigenvalues of H (b) Compute the eigenvectors of H, normalize them, and express them both as column vectors and in terms of | 1〉 and |2) (c) Denoting the two eigenvectors as lva) and |Vb), compute l/a) <>a and...
1. The aim of this problem set is to understand the dynamics of a spin-1/2 system in its full glory. Note that formally a spin-1/2 system and a qubit are equivalent hence, all what you will discover in this problem set will carry over to single qubits. Consider an electron spin (spin 1/2, magnetic moment gHB) interacting with a strong magnetic field Bo (0,0, B) in the z direction as well as with a much weaker magnetic field Brf =...
2. [10pts) In considering the EPR experiment, we used the anti-correlation property of the spin-0 state as an essential aspect of the analysis. In this question you are going to show that this property does not depend on the choice of direction of axis of the Stern-Gerlach analyzers. Show that the entangled state: |4= (1 131] 1)2 – 1 1/)2) is physically equivalent to the general state: 4) = - (11)in1)2n - | 1)in 1)2n) where ñ is a unit...
Question 1 (8 marks in total) The deuteron is a bound state of a proton and a neutron. Treating nucleons as identical particles with spin and isospin degrees of freedom, the total state of the deuteron can be writ- ten space Ψ spin Ψ isospin. The deuteron has a total angular momentum quantum number J - 1 and a total spin S -1. Our goal is to determine the parity of the deuteron Q1-1 (1 mark) Show that the possible...
A spin-3/2 system can have four values of angular momentum (Sz) when measured along the z- 2 2 If we measure the angular momentum of 1000 atoms, each prepared in the following state lp〉, how many times should we expect to measure an angular momentum of a) ? 2 Here 3 h the state of the system with an angular momentum of - etc.
4. 10 points The Spin operators for a spin-1/2 particle can be described by the Pauli matrices: 0 1 0 0 ,02= 0 -1 1 ¿ a) Write the normalized eigenvectors of Oz, I+) and 1-) which are defined such that 0z|+) = 1+) and 0z1-) = -1-), as column vectors in the same basis as the Pauli matrices given above. (You can assume without loss of generality that these eigenvectors are real.) (3 pts) b) Consider an eigenvector (V)...
lsa(1) lsB(1) 1Isa(2) 1sja 7. Consider this two-electron wave function: ψ-C Write the expression for ψ that comes from expanding the determinant. Find the normalization constant, C. The 1s orbitals are orthonormal, and so are the spin orbitals. a) b) Using your answer from (a), show that the wave function factors into a spin part and a spatial part. Hint: It may help to rewrite each spin orbit so its spatial and spin factors are clearer. For instance, rewrite Isa...