Question

2. (9 points total) Uncertainty relations. a) (1 point) Compute the commutator of the operators of coordinate and momentum in one dimension. b) (1 point) Two Hermitian operators A and B satisfy the relation [A, B] = iſ, where I is a number. Prove that I' is real. c) (1 point) Give the definition of the uncertainties A A and A B. d) (2 points) In this and subsequent parts of the question, we consider a normalized quantum stately) with zero average values of A and B. Use the Schwartz inequality to prove that AA AB >|(YAB Y). e) (1 point) Let C be a Hermitian operator. Prove that (y Cy) is real. Hint: all eigenvalues of C are real. f) (2 points) Use the results of parts b), d), and e) to prove the uncertainty relations for the operators, satisfying the conditions of part b). Hint: decompose AB = [(AB+BA)+(AB – BA)]/2 and prove that the expectation value of (AB + BA) is real. g) (1 point) A particle moves in one dimension. The uncertainty of its coordinate is 5 x 10-9 cm. Find the numerical value of the minimal possible uncertainty of its momentum.

Answer 1

(^) Intiaducing Let us calaulate [ *", Þ]g = x^ a test function gres ) i dz hese n=1 . [2.P] ith (b) (A8- BA) "= B*At- Atot [A,8] + At=A 4 8t=6 But for hermitian operator, [A,B]*= [BA-A@] (AG- GA) [A,B] %3D neue we can tee that it i auti hernutian. as : () The uncestainty in A LBs defined <A?> - <A)2 V<B2>- <8>2 and (4-CA) 14) & 1p>= (8-ce)t> (4) let x) = OA I4) the Sch warz inegnality is given as (x|x)<4l¢> > I<x1 +>}² Since A EB are hermitian hene OA deoB is also hemmiticn <x[x> = (410A}|4> <p,¢$> =<Y|eB;2/4> (x¢> = <4(oA O8)|4> sch warz inequality is giren as -

{8ABK41 ABIY> (<4I<14>j* = <41et/4> (e) 'all eigenvalue of hermitian operator is real inner pro duet of dwo wane always and ct=c also, functian ,ie, <414> is also rlal. is a mumber which <41ct/4>= <41e145 = real from gro, (f) BA. O8] + [aA,d8]={[A,8]+? [BA, 08j Since last term is posttive real no., e can h fer that or this is the saquired umcestainty relation. (9) uncentainty product ox. OPz> > -9 ox= 5X10 Cm = 5Xo" m -11 -34 6.626X10 4X3.14 x Sx j5" = 1.oS X10** kgms" -23 = 105 XID