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DEPARTMENT OF PHYSICSINDIAN INSTITUTE OF TECHNOLOGY, MADRASPH5100 Quantum Mechanics - 1 Assignment 1(30.7.2014) To be discussed on: 7.8.2014 One Dimensional Problems in Quantum Mechanics 1. At time t = 0 the state of a free particle is speciﬁed by a wavefunctionΨ( x, 0) =  Ae − x 2 a 2  + ik 0 x . Find the factor A and the region where the par-ticle is localized. Determine the probabilty current density J. Find thefourier transform of the wavefunction and the width of the wavepacketin k-space.2. Normalize the wavefunction Ψ =  e −| x | sinαx .3. Find the eigenfunctions and eigenvalues for sin  ddφ 4. Find the wavefunction and the allowed energy levels for a particle in apotential ﬁeld V(x) of the form V   ( x ) = 0 for 0 ≤ x ≤ a, 0 ≤ y  ≤ b, 0 ≤ z   ≤ c, =  ∞  for  x <  0 ,x > a,y <  0 ,y > b,z <  0 ,z > c 5. Consider the one-dimensional time-independent Schrodinger equationfor some arbitrary potential V(x). Prove that if a solution Ψ( x ) hasthe property that Ψ( x )  →  0 as  x  → ±∞ , then the solution must benon-degenrate and therefore real, apart from a possible overall phasefactor.6. Consider the one-dimensional problem of a particle of mass m in a po-tential V   = ∞ ,x <  0;V = 0, 0 ≤ x ≤ a ;V = V 0 ,x > a. (a) Show that the bound state energies (  E < V  0 ) are given by the equa-tion tan √  2 mEa    = −    E V   0 − E  . (b) Without solving any further, sketch theground state wavefunction.1  7. A particle of mass m moves non-relativistically in one dimension in apotential given by  V   ( x ) =  − aδ  ( x ). The particle is bound. Find thevalue of   x 0  such that the probability of ﬁnding the particle  | x | < x 0  isexactly equal to 1/2.8. A particle of mass m moving in one dimension is conﬁned to the region0  < x < L  by an inﬁnite square well potential. In addition, the particleexperiences a delta function potential  V   =  λδ  ( x  −  L/ 2). Find thetranscendental equation for the energy eigenvalues E in terms of themass m, the potential strength  λ , and the size L of the system.9. A particle, moving in one dimension, has a ground state function (notnormalized and do not normalize) given by Ψ 0 ( x ) =  e − α 4 x 44  (Where α  is a real constant) belonging to the energy eigenvalue  E  0  =    2 α 2 m  .Determine the potential in which the particle moves.10. A particle moves in a potential, in one dimension, of the formV(x) =  ∞  if   x 2 > a 2 else V(x) =  γδ  ( x ), where  γ >  0. For suﬃcientlylarge  γ  , calculate the time required for the particle to tunnel from beingin the ground state of the well extending from x = -a to x = 0 to theground state of the well extending from x=0 to x=a.2

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