quantum mechanics i lecture notes – graduate course...

Quantum Mechanics I

Lecture Notes – Graduate Course

UFRJ

Raimundo Rocha dos Santos

Friday 15

thAugust, 2014

2

Preface

Recommended literature:

• B = Gordon Baym, Lectures on Quantum Mechanics, (Westview, 1990).

• BD = Jean-Louis Basdevant and Jean Dalibard, Quantum Mechanics, (Springer,2002)

• CT = C Cohen-Tannoudji, et al.

• G = K Gottfried (1966)

• MZ = E Merzbacher

3

4

Contents

1 Fundamental Concepts 71.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.2 Wave functions and the Schrodinger equation. . . . . . . . . . . . . . . . . 71.3 Free particle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.3.1 �-function normalization . . . . . . . . . . . . . . . . . . . . . . . . 101.3.2 Periodic Boundary Conditions (PBC’s) . . . . . . . . . . . . . . . 111.3.3 Free wave packets . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.4 Heisenberg uncertainty relations . . . . . . . . . . . . . . . . . . . . . . . 141.5 Generalization to higher dimensions . . . . . . . . . . . . . . . . . . . . . 151.6 Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161.7 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2 Mathematical Framework 192.1 The space of wave functions . . . . . . . . . . . . . . . . . . . . . . . . . 192.2 State Space: Dirac notation . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.2.1 Kets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252.2.2 Inner product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3 The Postulates of Quantum Mechanics 27

4 Quantum Dynamics 29

5 Spin-1/2 and Two-level Systems 31

6 The Harmonic Oscillator 33

7 Angular Momentum in Quantum Mechanics 357.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357.2 Angular momentum algebra . . . . . . . . . . . . . . . . . . . . . . . . . . 36

7.2.1 Orbital Angular Momentum . . . . . . . . . . . . . . . . . . . . . . 367.2.2 Generalization: definition of Angular Momentum . . . . . . . . . . 36

7.3 General Theory of Angular Momentum . . . . . . . . . . . . . . . . . . . 367.4 Application: Orbital Angular Momentum . . . . . . . . . . . . . . . . . . 36

7.4.1 Eigenvalues and Eigenfunctions of L2 and Lz

. . . . . . . . . . . . 36

5

6 CONTENTS

7.4.2 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 367.5 Addition of Angular Momentum . . . . . . . . . . . . . . . . . . . . . . . 36

7.5.1 The Addition Problem . . . . . . . . . . . . . . . . . . . . . . . . . 367.5.2 Addition of Two Angular Momenta . . . . . . . . . . . . . . . . . 367.5.3 An Example: Two Interacting Spin-1/2 . . . . . . . . . . . . . . . 37

8 Symmetries 398.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

8.1.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 398.2 Transformacoes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

8.2.1 Transformacoes de Coordenadas . . . . . . . . . . . . . . . . . . . 408.2.2 Transformacoes de Estados Quanticos . . . . . . . . . . . . . . . . 418.2.3 Transformacoes de Observaveis . . . . . . . . . . . . . . . . . . . . 42

8.3 Deslocamentos no tempo . . . . . . . . . . . . . . . . . . . . . . . . . . . . 448.4 Translacoes Espaciais . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 468.5 Rotations and Angular Momentum . . . . . . . . . . . . . . . . . . . . . . 49

8.5.1 Rotations in 3D . . . . . . . . . . . . . . . . . . . . . . . . . . . . 498.5.2 Rotations in Hilbert Space . . . . . . . . . . . . . . . . . . . . . . 498.5.3 Rotations of Observables . . . . . . . . . . . . . . . . . . . . . . . 498.5.4 Rotation of States and Representations of the Rotation Operator . 508.5.5 Rotational Invariance and Conservation of Angular Momentum . 548.5.6 Tensor Operators and the Wigner-Eckart theorem . . . . . . . . . 55

8.6 8.6. Reflexoes Espaciais . . . . . . . . . . . . . . . . . . . . . . . . . . . . 598.7 8.7. Inversao Temporal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 598.8 8.8. Partıculas Indistinguıveis: Spin e Estatıstica . . . . . . . . . . . . . . 598.9 8.9. Aplicacoes de Teoria de Grupos . . . . . . . . . . . . . . . . . . . . . 59

8.9.1 8.9.1 Introducao . . . . . . . . . . . . . . . . . . . . . . . . . . . . 598.9.2 8.9.2. Exemplos de Grupos Finitos . . . . . . . . . . . . . . . . . . 59

9 Simetrias II 639.1 Introducao . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

9.1.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 639.2 Transformacoes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 639.3 Teste . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

Chapter 1

Review of Fundamental Conceptsin Quantum Mechanics

Refs.: CT, G, M

1.1 Introduction

We assume a prior knowledge of Quantum Mechanics (QM) at introductory level. Topicssuch as the experiments leading to the early foundations of QM, which highlight thewave-particle duality, are also assumed known; see the recommended literature.

The wave-particle duality is summarised by the de Broglie relations,

E = ~! (1.1.1)

p = ~k, (1.1.2)

where the particle-like variables, energy E and momentum p, on the left-hand sideare linearly related to the wave-like variables, frequency ! and wavevector k, throughPlanck’s constant, ~. The de Broglie wavelength, � = 2⇡/|k| = h/p, for a dust particle(m ⇠ 10�15 kg, and v ⇠ 1 mm/s), for a thermal neutron at room temperature, and forelectrons accelerated through a potential di↵erence of several hundreds of volts, can beestimated as � ⇠ 10�5A, 1.4 A, and 1 A, respectively. The smallness of the wavelengthalready for ‘macroscopic’ particles such as dust indicates that the wavelike properties canhardly be noticed in this case; by contrast, the wavelength of the order of interatomicdistances in a crystal shows that one can expect manifestation of wavelike phenomenasuch as di↵raction for neutrons and electrons.

1.2 Wave functions and the Schrodinger equation.

As a consequence of the de Broglie’s hypotheses, one arrives at the following formulationfor the quantum mechanical description of a material particle:

(i) The classical notion of a trajectory is replaced by the concept of a time-varyingstate, which, in turn, can be cast into correspondence with a complex wave function (r, t) containing all possible information about the particle.

7

8 CHAPTER 1. FUNDAMENTAL CONCEPTS

(ii) (r, t) is interpreted as a probability amplitude, in the sense that the probabilityof finding the particle, at time t, in the volume element d3r ⌘ dx dy dz centred atr is given by

dP(r, t) = | (r, t)|2 d3r, (1.2.1)

so that | (r, t)|2 ⌘ ⇤(r, t) (r, t) (⇤ denotes complex conjugation) becomes aprobability density, often assumed normalized, i.e.,

Z

all spacedP(r, t) =

Z

all spaced3r | (r, t)|2 = 1. (1.2.2)

Therefore, one must deal with square-integrable wave functions to represent phys-ically acceptable states.

(iii) The measurement of any physical quantity A is subject to the principle of spectraldecomposition; that is,

• The outcome must belong to a set of eigenvalues {a}.• Each eigenvalue a is associated with an eigenstate represented by an eigen-

function a

(r); this function is such that if (r, t0) = a

(r), where t0 is theinstant of time at which A is measured, then the measurement will alwaysyield a.

• For an arbitrary (r, t), the probability of finding the eigenvalue a at time t0can be obtained by first decomposing (r, t0) in terms of the set {

a

(r)},

(r, t0) =X

a

ca

a

(r), (1.2.3)

in terms of whose coe�cients the probability is given by

Pa

=|ca

|2Pa

|ca

|2 , (1.2.4)

where the denominator guarantees normalization, i.e., thatP

a

Pa

= 1.

• Collapse of the wave function: If the measurement of A in (r, t) at time t0yields a, then the wave function immediately after the measurement becomes

(r, t+0 ) = a

(r). (1.2.5)

(iv) (r, t) satisfies a wave equation, postulated by Schrodinger,

i~ @ (r, t)@t

= � ~22m

r2 (r, t) + V (r, t) (r, t), (1.2.6)

where m is the mass of the particle, r2 is the Laplacian operator, and V (r, t) isthe potential energy (or, simply, the potential). Several remarks are in order:

1.2. WAVE FUNCTIONS AND THE SCHRODINGER EQUATION. 9

• Classically, the state of a particle at a time t is characterized by six param-eters, the three position coordinates and the three velocity components. Bycontrast, the quantum state is characterized by an infinite number of param-eters, namely, the values (r, t) assumes on every point r. Therefore, theclassical trajectory is replaced by a wave propagation.

• The Schrodinger equation is linear and homogeneous in . As a consequence,if 1 and 2 are solutions, so is any linear combination, or superposition;therefore, the probability density displays wave-like interference e↵ects.

• The Schrodinger equation is first order in t. Therefore, the knowledge of (r, t0) determines the wave function in subsequent times.

• The Schrodinger equation does not admit creation or destruction of materialparticles; therefore, if a state is normalized in a given instant, it remainsnormalized at all times. Indeed, with the probability density,

⇢(r, t) ⌘ | (r, t)|2, (1.2.7)

and the probability current density,

j(r, t) ⌘ ~2mi

[ ⇤(r, t)r (r, t)� (r, t)r ⇤(r, t)] , (1.2.8)

conservation of probability is expressed in the form of a continuity equation,

@⇢

@t+r · j = 0. (1.2.9)

[The reader should derive this result. Do you have to impose any restrictionson V (r, t) to arrive at the continuity equation?]

Integrating the continuity equation in all space, and assuming j decreasesfaster than 1/r2 for r ! 1, we obtain

@

@t

Zd3r ⇢(r, t) = 0, (1.2.10)

which shows that the wave equation guarantees the conservation of normal-ization.

(v) For time-independent potentials, the Schrodinger equation is separable: seek solu-tions of the form

(r, t) = (r) e�iEt/~, (1.2.11)

where (r) and E ⌘ ~! are determined from the time-independent Schrodingerequation,

H (r) = E (r), (1.2.12)

which, with the identification p ! �i~r, allows us to interpret H as the Hamilto-nian operator,

H ⌘ � ~22m

r2 + V (r), (1.2.13)

10 CHAPTER 1. FUNDAMENTAL CONCEPTS

whose eigenvalue E is the total energy.

Quantum states in the form (1.2.11) are called stationary : | |2 does not dependon t. This connects with Bohr’s early postulate that an electron in a stationaryorbit of the hydrogen atom does not radiate.

1.3 Free particle

Consider free one-dimensional motion. The Schrodinger equation becomes

i~ @ (x, t)@t

= � ~22m

@2 (x, t)

@x2, (1.3.1)

with solutions

motion along + x : (x, t) = Aei(kx�!t) ) | |2 = |A|2 (1.3.2)

motion along� x : (x, t) = Aei(�kx�!t) ) | |2 = |A|2, (1.3.3)

where A is a normalization constant.In either case, not square-integrable functions: this inconsistency reflects the fact that

a particle cannot occupy the whole space; it is an idealization as much as electromagneticplane waves are.

Patching up:

1.3.1 �-function normalization

We use the representation of the �-function,

Z 1

�1dx e�ikxeik

0x = 2⇡�(k � k0), (1.3.4)

to impose Z 1

�1dx ⇤

k

0(x) k

(x) = �(k � k0), (1.3.5)

where the subscript k has been used to stress that the wavevector parameterizes thewave function,

k

(x) = Aeikx (1.3.6)

With this,

A =1p2⇡

. (1.3.7)

Clearly this procedure does not turn the plane wave into a square-integrable function;it just casts the divergence into a ‘manageable one’, which is important for the internalconsistency of the Quantum Mechanics formalism.

1.3. FREE PARTICLE 11

1.3.2 Periodic Boundary Conditions (PBC’s)

(also known as ‘box-normalization’ )We impose

(x+ L, t) = (x, t) (1.3.8)

where L is a constant, representing the (finite) size of the system, which becomes

Aei(kx+kL�!t) = Aei(kx�!t) ) eikL = 1. (1.3.9)

With this, the k-values are quantized,

k =2n⇡

L, n = 0,±1,±2, . . . (1.3.10)

That is, one must have an integer number of wavelenghts in the box. Notice that thisis not the same as imposing = 0 at the edges of the box, as if there were an infinitepotential outside the box; in this case, the quantization condition would be to fit aninteger number of half -wavelengths in the box.

Given that the particle must be found within the box, the normalization conditionbecomes Z

L

0dx | (x, t)|2 = 1 ) A =

1pL. (1.3.11)

This normalization therefore explores the fact that the space occupied by the particleis actually limited, not infinite.

1.3.3 Free wave packets

Assume the particle is confined in the form of a wave packet; as we will see below, thislocalization provides a natural way of connecting with the classical motion. The way toachieve the sought confinement is known since the works of Joseph Fourier in the 19thCentury: use plane waves as basis functions, with a suitable choice of coe�cients.

Specifically, since k is a continuous variable, we expand the wave function at t = 0as an integral over the plane waves,

(x, 0) =1p2⇡

Z 1

�1dk �(k) eikx, (1.3.12)

whose coe�cients, �(k), are determined by the inverse relation,

�(k) =1p2⇡

Z 1

�1dx (x, 0) e�ikx. (1.3.13)

Since the plane waves propagate in time independently of each other, we may write

(x, t) =1p2⇡

Z 1

�1dk �(k) ei[kx�!(k)t], (1.3.14)

12 CHAPTER 1. FUNDAMENTAL CONCEPTS

Figure 1.1: (a) One-dimensional momentum distribution centred at k0. (b) Evolution ofthe wave packet during a small time interval. (Extracted from BD, Fig. 2.2).

where, for future use, we write !(k), though for a free particle ! = ~k2/2m, by virtueof Eqs. (1.1.1) and (1.1.2).

If we take �(k) = �(k � k0), a plane wave is recovered, (x, t) = Aei[k0x�!(k0)t]. Letus now provide some width to �(k), assuming it is symmetrically distributed about somevalue, k0, with width �k [measured at one half of the maximum of �(k)] such as the onedepicted in Fig. 1.1. Note that (�k)�1 provides a natural length scale.

We write�(k) = |�(k)|ei↵(k), (1.3.15)

and assume ↵(k) is smooth where |�(k)| is appreciable (i.e., within an interval on theorder of �k around k0), so that it can be expanded as

↵(k) ' ↵(k0) + (k � k0)x0, (1.3.16)

where

x0 ⌘ �d↵

dk

���k=k0

(1.3.17)

has units of length. Equation (1.3.12) can then be written as

(x, 0) ' ei[k0x+↵(k0)]

p2⇡

Z 1

�1dk |�(k)|ei(k�k0)(x�x0), (1.3.18)

which is useful to draw interesting conclusions.For |x� x0| � (�k)�1, the integrand oscillates very rapidly within the interval �k,

leading to a vanishingly small integral (i.e., the plane waves interfere destructively): theprobability of finding the particle at points x � x0 is negligible. On the other hand, for|x� x0| ⌧ (�k)�1 the integrand hardly oscillates; for x ' x0, in particular, | (x, 0)| ismaximum (now the plane waves interfere constructively). In other words, x0 marks theposition, x

M

, where the overall phase in (1.3.12),

�(k, x) ⌘ kx+ ↵(k), (1.3.19)

is stationary,

@�

@k

���k=k0

= x+d↵

dk

���k=k0

= 0 ) xM

= x0 = �d↵

dk

���k=k0

. (1.3.20)

1.3. FREE PARTICLE 13

When |x � x0| & (�k)�1 the integrand oscillates at least once, and we can take|x� x0| ⇠ �x, a measure of the width of the packet, to establish

�x�k & 1. (1.3.21)

This result recovers what is known from Fourier analysis: to build more localized packets(smaller width �x), one has to include a larger number of Fourier components, hence alarger width in momentum space, �k; and vice-versa.

We now discuss the time evolution of the wave packet.

(i) Maximum of the packet.

Starting from Eq. (1.3.14), and incorporating the phase of �(k) in the exponent,we get

(x, t) =1p2⇡

Z 1

�1dk |�(k)| ei�(k;x,t), (1.3.22)

where�(k;x, t) = kx� !(k)t+ ↵(k). (1.3.23)

The condition of stationary phase, @�/@k = 0, now yields

xM

(t) = �d↵

dk

���k=k0

+d!

dk

���k=k0

t = xM

(0) + vg0t, (1.3.24)

where vg0 ⌘ v

g

(k0) is the group velocity, vg

(k) ⌘ d!/dk, evaluated at k = k0. Theposition of the maximum of the wave function therefore follows the motion of aclassical free particle with velocity v = v

g

= ~k0/m; notice that the phase velocity,vph

= !/k, is one half of the group velocity.

(ii) Form of the packet.

For a momentum distribution sharply peaked at k0, we may expand !(k) in powersof (k � k0); see Fig. 1.1(a). Keeping only terms to first order, i.e.,

!(k) = !0 + vg0(k � k0) +O[(k � k0)

2], (1.3.25)

where !0 ⌘ !(k0), Eq. (1.3.14) becomes

(x, t) =1p2⇡

Z 1

�1dk �(k) ei[kx�!0t�vg0(k�k0)t]

=ei!0t

p2⇡

Z 1

�1dk �(k)eik(x�vg0t)

= ei!0t (x� vg0t, 0), (1.3.26)

that is, the packet moves without changing its shape,

| (x, t)|2 = | (x� vg0t, 0)|2. (1.3.27)

14 CHAPTER 1. FUNDAMENTAL CONCEPTS

Certainly this only occurred because the expansion in Eq. (1.3.25) did not includethe quadratic term; this, in turn, is only justifiable if

1

2

d2!

dk2

���k=k0

(k � k0)2t ⌧ 1 ) ~t

2m(�k)2 ⌧ 1. (1.3.28)

Therefore, the packet expands for longer times; how much longer depends on themomentum width, and on the mass of the particle: packets representing lightparticles deform faster than those corresponding to heavy particles.

As a specific example, in Problem 4 one shows that when the momentum distribu-tion, �(k), is real, and the centre of the packet is initially at the origin, the spreadis governed by

(�x) = (�x)0

s

1 +~2m2

(�k)2

(�x)20t2, (1.3.29)

irrespective of its shape, where (�x)0 is the initial width of the packet.

1.4 Heisenberg uncertainty relations

Using the de Broglie relation, Eq. (1.1.2), we can write Eq. (1.3.21) as

�x�p & ~; (1.4.1)

this is, apart from a factor of 1/2 on the right hand side, the famous Heisenberg uncer-tainty relation.

We can ascribe a statistical interpretation to the quantities on the left-hand sideof Eq. (1.4.1); for simplicity, we still restrict our discussion to one-dimensional systems.Imagine we identically prepare (from the macroscopic point of view) a large number offree single-particle systems. In each system, we perform simultaneous measurements ofposition and momentum of the particle; the data will certainly show some scattering,reflecting the uncertainty in the measurements. With the data at hand, we calculate theaverages hxi and hpi, together with their corresponding standard deviations,

�x =p

hx2i � hxi2, (1.4.2)

and�p =

php2i � hpi2. (1.4.3)

Then, Eq. (1.4.1) simply states that the uncertainties in these measurements, given bythe above standard deviations, are not independent: if the precision in position mea-surements increases (i.e., smaller �x), a wider scattering in momentum data ensues, andvice-versa. It is important to keep in mind that the bounds imposed by the Heisenberguncertainty relations should not be attributed to inherent limitations of the measure-ment apparatus, but as intrinsic manifestations of the ultimate quantum nature of thesystem. Later on (Sec. X.Y), we will provide a more general proof of the Heisenberguncertainty relations, with the aid of the concept of incompatible measurements.

1.5. GENERALIZATION TO HIGHER DIMENSIONS 15

Fourier analyses of time and frequency also lead to relations between frequency andtime

�⌫�t ⇠ 1, (1.4.4)

where �⌫ is a frequency range and �t a time interval; for instance, to produce a pulselasting 1 µ s, one needs to combine waves within a bandwidth (frequency range) of theorder of MHz. Using the de Broglie relation (1.1.1), ignoring the factor 2⇡ relating ⌫and the angular frequency !, and changing the relation to a lower bound leads to

�E�t & ~. (1.4.5)

In relation to a wave packet, we interpret �E as the standard deviation in energymeasurements, and �t represents an intrinsic time scale of the system, not of the mea-surement apparatus: it can be the time the packet takes to pass one point in space[i.e., �t ⇠ m�x/(~k0), k0 is the centre of the momentum distribution], the lifetime of aparticle, the inverse of some natural frequency of the system, and so forth.

The uncertainty relations reflect what is known as quantum fluctuations: particlesare in motion even in their lowest state. For instance, while the state of lowest energyin a classical oscillator of natural frequency ! corresponds to a particle at rest at thepoint of equilibrium, the lowest possible energy in the corresponding quantum oscillatoris (d/2)~!, where d is the dimensionality of the system; that is, the quantum particle isin motion.

1.5 Generalization to higher dimensions

Most of the discussions in Sections 1.3 and 1.4 have focused on the one-dimensional case.Though in most cases the generalisation to higher dimensions is trivial, here we mentionsome of them, for the sake of completeness. The three-dimensional initial wave packetis now written in terms of plane waves as

(r, 0) =1

(2⇡)3/2

Zd3k �(k) eik·r, (1.5.1)

with

�(k) =1

(2⇡)3/2

Zd3r (r, 0) e�ik·r, (1.5.2)

so that

(r, t) =1

(2⇡)3/2

Zd3k �(k) ei[k·r�!(k)t], (1.5.3)

where all integrals are understood to be over the whole space (position or reciprocal, asapplicable).

The uncertainty relations between like-components of position and momentum alsofollow,

�x�px

& ~, �y�py

& ~, and �z�pz

& ~, (1.5.4)

but, as we will see later, there is no uncertainty relation between di↵erent components,such as �x�p

y

, and so forth.

16 CHAPTER 1. FUNDAMENTAL CONCEPTS

1.6 Further Reading

• Order of magnitude estimates: CT Complement AI.

• Uncertainty relations and atomic parameters: CT Complement CI.

• Relation between one- and three-dimensional problems: CT Complement FI.

• One-dimensional Gaussian wave packets: CT Complement GI.

• One-dimensional square potentials: CT Complement HI.

• Wave packets at a potential step: CT Complement JI.

1.7 Problems

1. Show that in one-dimensional problems the energy spectrum of bound states is alwaysnon-degenerate.

2. Let (r, t) be an eigenfunction of the Schrodinger equation corresponding to theenergy E. We will see later that the time reversed (r, t) is obtained by taking botht ! �t and the complex conjugate of (r, t).

(a) Show that the function ⇤(r,�t) satisfies the same Schrodinger equation as (r, t), with the same energy E.

(b) Consider a stationary solution, E

(r) e�iEt/~. Show that if the eigenvalue E isnon-degenerate, then

E

(r) is real, apart from an overall constant and arbitrarycomplex factor.

3. Obtain the wave functions and eigenenergies for a particle of mass m in a one-dimensional box, i.e.,

V =

(0 for 0 x L

1 otherwise.

Compare with the results for the case of periodic boundary conditions [i.e., V = 0, 8x,but (x+ L) = (x)].

4. Solve the time-independent Schrodinger equation in one dimension for the potentialV (x) = � �(x). Consider the repulsive and attractive cases. For the attractive case,compare your results with the solutions for a square well in the limit V0 ! 1 anda ! 0, with V0a finite (see, e.g., CT, Complement HI). For both attractive andrepulsive cases, calculate the transmission coe�cient for positive energies.

5. Mostre que se a distribuicao de momentos para um pacote livre, �(p), e real e aorigem e escolhida de modo que, inicialmente, hxi = 0, entao a equacao

(�x)2 = (�x)2|t=0 +

(�p)2t2

m2

1.7. PROBLEMS 17

e verdadeira para um pacote com forma arbitraria. (Sugestao: use a representacaodos momentos). Note a dependencia da largura do pacote com a massa da partıcula!

6. Considere um pacote de onda gaussiano que, em t = 0, seja dado por

(x, 0) =1

(�2⇡)1/4e�x

2/2�2

eik0x,

onde � e k0 sao constantes.

(a) Mostre que a distribuicao de momentos tambem e gaussiana;

(b) Mostre que este pacote corresponde a menor incerteza possıvel: �x�p = ~/2;(c) Mostre que a densidade de corrente para t = 0 e dada por j(x, 0) = ⇢v0, onde

⇢(x, 0) = | (x, 0)|2 e v0 = ~k0/m.

(d) Calcule a funcao de onda para t > 0 e mostre explicitamente que o pacote livrese alarga com o tempo de acordo com

�x(t) =�p2

r1 +

~2t2�4m2

.

Verifique que seu resultado se reduz ao do Problema 5.

(e) Calcule a densidade de corrente para t > 0, e compare com o resultado obtidoem (c); verifique o que ocorre no maximo do pacote, x0 = v0t.