Example of using the eigenvalues to classify states: Write down the differential equation for L2 (the total angular mom...
Rotational states of a diatomic molecule can be approximated by those of a rigid rotor. The hamiltonian of a rigid rotor is given by hrotor 12/21, where L2 is the operator for square of angular momentum and I the moment of inertia. The eigenvalues and eigenfunctions of L2 are known: Lylnu =t(1+1)ay," , where m.--1, , +1 a) Calculate the canonical partition function : of a rigid rotor. Hint: Replace summation over by integral. b) What is the probability that...
2. The hydrogen atom [8 marks] The time-independent Schrödinger equation for the hydrogen atom in the spherical coordinate representation is where ao-top- 0.5298 10-10rn is the Bohr radius, and μ is the electon-proton reduced mass. Here, the square of the angular momentum operator L2 in the spherical coordinate representation is given by: 2 (2.2) sin θー sin θ 00 The form of the Schrödinger equation means that all energy eigenstates separate into radial and angular motion, and we can write...
1) Write the following wave functions of the Hydrogen atom b100(r, 0, )= 1s b200(r, 0, ) 2s; b21+1(r,0, )= b2p 2) Calculate the medium radius and possible radius for this functions. 3) What are the energy at each state? 4) Calculate the angular momentum of each state using the differential operator 1 L2 h2 1 sin sin2 0 2 sin e ae 5) Verify the above results with the equation L2nlm = l( 1)h2bn{m 6) Calculate the components L2...
Exercise 1: The helium atom and spin operators 26 pts (a) Show that the expectation value of the Hamiltonian in the (sa)'(2a)' excited state of helium is given by E = $42.0) (Avo ) anordes ++f63,(-) (%13-12 r) 62(e)drz + løn.(r.) per 142, (ra)]" drų dr2 - / 01.(ru) . (ra) Anemia 02.(r.)61.(r.)dr; dr2 (1) Use the approximate, antisymmetrized triplet state wave function for the (Isa)'(280)' state as discussed in class. Hint: make use of the orthonormality of the hydrogenic...
2. Addition of Angular Momentum a) (8pts) Given two spin 1/2 particles, what are the four possibilities for their spin configuration? Put your answer in terms of states such as | 11). where the first arrow denotes the z-component of the particle's spin. Identify the m values for each state. b)(7pts) If you apply the lowering operator to a state you get Apply the two-state lowering operator S--S(,) +S(), where sti) acts on the first state and S acts on...
Question 8 please 5. We start with Schrodinger's Equation in 2(x,t) = H¥(x,t). We can write the time derivative as 2.4(x, t) = V(x,+) - (xt), where At is a sufficiently small increment of time. Plug the algebraic form of the derivative into Schrodinger's Eq. and solve for '(x,t+At). b. Put your answer in the form (x,t+At) = T '(x,t). c. What physically does the operator T do to the function '(x,t)? d. Deduce an expression for '(x,t+24t), in terms...
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...
where lo, l1 are the angular momentum quantum numbers for the lower and upper states respectively of the atomic transition (s,p, d,...or 1-0, 1, 2, . . .). For example, for the 4s-3p transition for sodium (see Fig. 1 below) has n0-3, nı = 4, 10-1, 11-0 Transition levels of Sodium 6s 4.0 d series s series 2. Ep-3 p series Figure T: Transition levels for the neutral sodium atom The 3p-3s transition produces strong yellow emission spectral lines called...
Exercise 1 (a) Write down the general form of the Clebsch-Gordan decomposition of states lji,j2;j, m) into l/i,j2;m,m2) and vice versa. Specify explicitly the summations by indicating the variables that are summed over and their ranges (b) Use the table below to write down the Clebsch-Gordan decompositions of the states 1, m)ll3, m2)1l1,-1) and l,ml, m2)1,0)1,-1), and relate these two decompositions by usingL on the first of these states. 34. CLEBSCH-GOR DAN COEFFICIENTS, SPHERICAL HARMONICS, AND d FUNCTIONS Notatiorn :|...
A NON stationary state A particle of mass m is in an infinite square well potential of width L, as in McIntyre's section 5.4. Suppose we have an initial state vector lv(t -0) results from Mclntrye without re-deriving them, and you may use a computer for your math as long as you include your code in your solution A(3E1) 4iE2)). You may use E. (4 pts) Use a computer to plot this probability density at 4 times: t 0, t2...