Download Quantum Mechanics Problems and more Exams Quantum Mechanics in PDF only on Docsity!
1. [4 rnarks] Consider a particle of mass rn, moving inonedimension, unde.r the inftucricr.
of a potential V(x ) given by
V (:x: )==--^ k ~
I :x: 1.
wh ere k is a constant.. (i) Estimate the ground state energy ofthe particle using the uncertainty principle.
(ii) How does the ground state energy change if we let h -t 0.
2. [6 rnarks] Consider a particle of mass rn subj(~ct to a onc-dirrwnsional potential V( :1,:) given by
( oc,
V(x) ==
-o:5(x) 1
:r; < - -d
:r; > -d
where a and d arc positive constants and J(x) is the Dirac delta-function.
(i) Find an equation for the bound stateenergy E of the particle.
(ii) How docs the energy E change if vvc let d ·---, x.
1. f5 marks] Consider a particle of mass rn subject to thr potential V(.r) - = k:r'^1 with
k > 0. Assurnr. thi-lt. the eigenfunctions o!' the particle bc~bavc as e:Lp[ - -a.l:1,f '] for l:rl --> x,
vvh<'rc u and n are constants. Csc asymptotic analysis of thr. corresponding Schroedinger equation of the particle to find the values of a and n.
2. [4 marks] In one dirncnsion, a ket If) is defined through
where N; :r 0 and a are constants. Find (kif), where p = hk.
3. {E, marks] Cor..sidcr the dif;c~ctc orthonormal basis {lv, 1. ) }, 't = l , 2; ... j / that spans
nil N- - dinwnsional Hilbert space. Form a projection operator P vvhich docs not. includ e the'
(a) Is P 2 = P?
(b) Is P an i~entity operator?
(c) Compute the trace of P.
N
P = L lui) (u,i j
i=1,i:f
4. '[5 marks] Consider two Hermitian operators A and B, in a three-dimensional Hilb rrt
space, being represented by
a 0 0 b ()^0
A= (^) 0 -a (^) 0 B = (^0 0) ib 0 0 -a^ 0 -^ i.b^ 0
where a and b are real constants.
(a) Show that A and fl corn mut e.
(b) Find a set of orthonorrnal k<! t.s wl1ich an! sinntlLanc~uus c·i gc~ nk C'ts of both ; \ ~rnd /J.
1. [5 marks] Consider the ground state of a one dimensional simple harmonic oscillator.
Let X ( t) be the position operator of the oscillator in the Heisenberg representation. Evaluate
(X ( t )X (0)) in the ground state of the oscillator.
2. [5 marks] Consider a simple harmonic oscillator with its Hamiltonian H 0 given by
Ho==-+-^ p2^ x
where we have used a simplified notation and put n == m == w == l. Next, we add a term -J2a,X to H 0 so that the new Hamiltonian, H, becomes
H == -^ p2 + -x2 - \1'2a_X 2 2
where a is a real number. Let la) denote the ground state of the new Hamiltonian H. Using
creation and annihilation operators, show that
a la) == a la)
·where a is the annihilation operator corresponding to H 0.
1. [5 marks] Consider a linear harmonic oscillatorstate I¢), given by I¢) = a l</>o)^ +^ b^ 1¢1),
where /¢0 )^ and^ /¢1 )^ are^ the^ normalized , ground state and^ the^ first excited energy eigenstates of the linear harmonic oscillator in one dimension. The^ state^ 1¢)^ is^ also normalized.^ Find^ the values of a and b which will maximize^ (¢IX^ I¢),^ where^ X^ is the^ position operator.
2. [7 marks] Consider a system having a 3 x 3 Hilbert space.^ Let^ the^ three orthonormal, energy eigenstates ofthesystem be denoted by^ lm 1 ) ,^ lm 2 )^ and^ lm 3 ).^ Interestingly, in different nuclear reactions^ three^ different statesof^ the^ system, which are^ not^ the^ energy eigenstates, are produced. Let us^ denote^ these^ three^ orthonormal^ states^ as^ IJ 1 ),^ lh)^ and^ 1/3).^ If^ thestate of^ the system at time t = 0 is l'l/J(t = 0)) = Iii) then find the probability that the^ system^ remains in
\Ji) at some later time t > 0.
Take the matrix elements of (mJ Iii) to be
a b 0 -b a 0 0 0 1
with {i = 1, 2, 3} and {j = 1,2,3}. We also have a and basreal and time indepenednt^ constants
with a 2 +^ b^2 =^ 1.^ The^ eigenvalues of^ the^ energy eigenstates^ lmi)^ are given by^ mic^2 ,^ where^ m,i^ is
the mass and c isthe speed of light.
3. [7 marks] Consider a system with orbital angular momentum^ quantum^ number^ l.^ Let
lmi) with^ {i^ = 1,... , (2l^ +^ 1)}, be^ the^ eigenvectors of^ the^ z-component of^ the^ orbital angular
momentum operator £ 2 of the system. Using L+ and L,_ the raising and lowering operators^ of
the system respectively, show that (mdmJ) = 6 iJ·
4. [7 marks] Consider an electron in the ground^ state^ of a hydrogen-like system^ A,^ where^ t hC' nucleus has two neutrons in addition to one proton. System A undergoes nuclear react ion which changes one of the neutrons into a proton instantaneously. Let^ us^ denote the^ new,^ hydro^ ge^ n-like. system by B. Calculate the probability that the electron remains in the ground state of^ th<-'^ n^ ew system B.
