Joel Broida UCSD Fall 2009 Phys 130B QM II. Homework Set 2 DO ALL WORK BY HAND IN OTHER WORDS, DON T USE MATHEMAT- ICA OR ANY CALCULATORS.

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1 Joe Broida UCSD Fa 009 Phys 30B QM II Homework Set DO ALL WORK BY HAND IN OTHER WORDS, DON T USE MATHEMAT- ICA OR ANY CALCULATORS. You may need to use one or more of these: Y 0 0 = 4π Y 0 = 3 4π cos Y = 3 8π eiφ sin.. In the cass notes it is shown that the top orbita anguar momentum state is given by [ ] / ( + )! Y (, φ) = ( ) 4π! (sin ) e iφ := c (sin ) e iφ. In this probem you wi fi in the detais in the derivation of an arbitrary Y m (, φ) by appying the owering operator ( L = e iφ i cot ) φ m times to Y. (a) By appying L to Y, show that Y (, φ) = c e i( )φ sin cos. In principe, we can appy L repeatedy to generate a of the Y m, but this gets very messy. The easy way is based on part (b) beow. (b) Prove the foowing resut: ] L [e imφ f()] = e [sin i(m )φ m d d cos sinm f(). (Hint: You wi need to show df d cos = df sin d and d sin = cot.) d cos As a consequence, you have shown that ] L [e iφ sin ] = e [sin i( )φ d d cos sin sin = L Y c.

2 (c) Now show that (L ) Y = (L ) [e iφ sin ] c = e [sin i( )φ d ] d(cos) sin sin and hence aso that (using (L ) Y = 4 Y ) Y (, φ) = c ( ) ei( )φ sin d d(cos ) sin. The important point to reaize here is that the sin term in the midde is canceed when the operator L is appied twice. As you wi see next, this hods true for a higher powers of L. (d) To see what the genera resut is, appy L a third time to show (L ) 3 Y = (L ) 3 [e iφ sin ] c [ = 3 e i( 3)φ sin 3 d 3 ] d(cos ) 3 sin sin and hence aso show Y 3 = c ( )( )3 ei( 3)φ sin 3 d 3 d(cos) 3 sin (e) Show that the resut in part (d) can be generaized to Y m ( + m)! (, φ) = c ()!( m)! eimφ sin m d m d(cos ) m sin. This is just equation (.4) in the notes on Anguar Momentum.. In cass we defined anguar momentum in the cassica sense as L = r p, and then interpreting r and p as quantum mechanica operators, we were abe to derive the commutation reations [L i, L j ] = iε ijk L k. We then showed that rotating a (scaar) wave function ψ(x) eads to a rotated state ψ R (x) given by ψ R (x) = e (i/ ) L ψ(x). In the case of an infinitesima rotation, this can be written as ψ R (x) = [ (i/ ) L]ψ(x). This is why L is caed the generator of rotations. A more genera approach is to essentiay reverse this ogic. We have seen that spin one-haf partices are represented by a two-component spinor (or vector), because the z component of spin can take one of two vaues (±/).

3 A spin-one partice has three z-components of spin (0, ±) and woud be represented by a three-component vector. In other words, a partice with spin is represented by a muti-component vector. Furthermore, the usua approach in quantum mechanics is to start from a cassica observabe quantity and then construct the corresponding operator that acts on the Hibert space of states of the system. In this case, we start from a spatia rotation in three dimensions, and then construct the corresponding rotation operator. In other words, corresponding to an infinitesima rotation R(), we define the generator of rotations J by U(R()) = i J where U(R()) is the quantum mechanica operator that acts on the state ψ. In cass we saw that transations commute, i.e., that the transation group is abeian. However, rotations in three dimensions do not commute in genera, and the rotation group is not abeian. In this probem you wi derive the commutation reations of the generators J i using this fact. In three dimensions, the matrices that rotate a vector about the x, y and z axes are 0 0 R x () = 0 cos sin R y () = 0 sin cos cos 0 sin 0 0 sin 0 cos cos sin 0 R z () = sin cos (a) Let = ε and expand these to first order in ε to write each of them in the form R i (ε) = I + εg i i = (,, 3) = (x, y, z) where G i is a 3 3 matrix. (b) Show that 0 0 [R x (ε), R y (ε)] = ε

4 (c) Show that to order ε we can aso write R z (ε )R y (ε)r x (ε) = R x (ε)r y (ε). (d) Carrying the resut of part (c) over to quantum mechanics, we repace each rotation matrix by it s corresponding quantum mechanica operator U(R(ε)) = (i/ )ε J acting on an arbitrary state, and we can therefore write [ ( i )( εj x i ) εj y ( i )( ε J z i )( εj y i x) ] εj ψ Expand this to order ε and show that [J x, J y ] = i J z. = 0. By cycicay permuting the indices x y z x in part (c), it is not hard to show that in genera we have [J i, J j ] = i ε ijk J k and thus the generators of rotation are in fact the anguar momentum operators. 3. Find the matrix eement m L x m. 4. Let the state function of an eectron be ψ = R(r) [ 3 Y 0 (, φ)χ Y (, φ)χ ] where R(r) is a radia wave function and χ ± are the eigenfunctions of S z. (a) Show directy that the z-component of the eectron s tota anguar momentum is /, and that the eectron has orbita anguar momentum unity. (b) What is the probabiity density for finding the eectron with spin up at (r,, φ)? What about spin down? 5. Consider an anguar momentum system represented by the state vector ψ = What is the probabiity that a measurement of L x yieds the vaue 0? [Hint: There are at east two ways to do this probem, and neither of them requires a ot of agebra if you re a itte bit cever.] 4

5 6. Suppose the wave function of a partice in a sphericay symmetric potentia is ψ(x) = c(x + y + z)e (x +y +z )/α where c is a normaization constant. What is the probabiity that in a measurement one wi find L = and L z =? 7. A spin / partice is in an eigenstate of S x with eigenvaue + / at time t = 0. At that time it is paced in a magnetic fied B = (0, 0, B) in which it is aowed to precess for a time T. At that instant the magnetic fied is very rapidy rotated in the y-direction, so that its components are (0, B, 0). After another time T a measurement of S x is carried out. What is the probabiity that the vaue / wi be found? 8. Suppose two eectrons are in the singet state 0 0 = ( + + ). where + is shorthand notation for () etc. Let S a be the component of spin for partice in the direction â, and et S () b be the component of spin for partice in the direction ˆb. Show that the expectation vaue of the operator S a () S () b is given by S a () S() b = 4 cos where â ˆb = cos. (Hint: Since two vectors define a pane, you might as we et â = ẑ and et ˆb ie in the xz-pane.) 9. Construct a possibe tota spin states that resut from combining two spin partices. 0. An eectron in a hydrogen atom is in the combined spatia and spin state [ ] ψ = R (r) 3 Y 0 (, φ)χ Y (, φ)χ where the radia function is given by R (r) = a 3/ r 0 e r/a0 4 a 0 and is the Bohr radius. a 0 = m e e = 0.59 Å 5

6 (a) If you measured L, what vaues coud you get and with what probabiities? (b) What if you measured L z? (c) What if you measured S? (d) What if you measured S z? (e) What if you measured J? (f) What if you measured J z? (g) What is the probabiity density of finding the eectron at the position (r,, φ)? (h) Since the operator S z and the operator r commute, both can be simutaneousy measured. What is the probabiity density of finding the partice with spin up at the radius r? 6