Module 1 : Quantum Mechanics

Chapter 4 : Quantum mechanics in 1-dimension

Quantum mechanics in 1-dimension

We will now apply the concepts of quantum mechanics to analyse some specific problems. We begin withproblems in 1-dimension.

4.1 Free particle and wave packet

The Hamiltonian for a free particle in 1-dimension is

(4.1)

The relevant observables are and or . It is interesting to note that parity which inverts the

coordinates is an important observable. Defining parity operator as

(4.2)

Also which implies that the eigenvalues of are corresponding to even or odd wave functions. Now

since

(4.3)

we can have simultaneous eigenfunctions of energy and momentum, or energy and parity, but not ofmomentum and parity. We begin with the Schroedinger equation

(4.4)

We consider special, separable solutions which lead to

(4.5)

which can be satisfied only if both sides are constant, say . This leads to

(4.6)

The solutions to these equations are

(4.7)

These separable solutions are eigenfunctions of energy with eigenvalues . Some significant points ofthese solutions need to be noted. 1. The solutions are not normalizable in the usual way. However, if we take , we can regard it as an

eigenfunction of , and get

(4.8)

We take for -function normalization.

2. For each given , there are two degenerate states with the same energy. Now since commutes with and

, we can have simultaneous eigenstates of and , or of and . These are

(4.9)

3. Though the solution is not normalizable, it can be used for describing relative probabilities. One may also use

these solutions to describe a beam of particles with representing the density of particles per unit length. We

have for the flux of particles,

(4.10)

For the standing waves, the flux is zero, e.g. for eigenstates of . 4. The most general solution is

(4.11)

This satisfies the Schroedinger equation but in general is not an eigenfunction of energy, or momentum, or parity.However it can be normalized and can be used to describe a free particle. The normalization condition implies

We have already considered as a wave function in the momentum space, Eq.(3.20). A general solution of this

type is called a wave packet.The implications of a wave packet are understood by considering a narrow distribution

for around . Expanding the exponent in Eq.(4.11) around and retaining the first two terms,

one gets

(4.13)

Here, is called the group velocity in contrast to which is called the phase velocity,

. The group velocity simulates classical motion. The localization of the particle is

determined essentially by . Consider a simple example,

(4.14)

It is normalized. Then is

(4.15)

It is peaked around and the peak is spread over . Together with the spread in the

momentum, one has

(4.16)

(4.17)

Normalization condition implies

(4.18)

The time-dependent wave function is

(4.19)

Using one obtains

(4.20)

This is the wave packet which spreads,

(4.21)

which has the smallest uncertainty at .

4.2 Uncertainty principle

A rigorous statement of the uncertainty principle can be made as follows. Consider

(4.22)

where and are hermitian operators, and are their average values, and

and are their standard deviations. Then one has

(4.23)

Proof: With , we have

(4.24)

Since and are hermitian, this leads to

(4.25)

Minimizing with respect to and , one gets

(4.26)

Using these in Eq.(4.25) leads to

(4.27)

where we have used the relation

(4.28)

With

(4.29)

where and are hermitian operators, one obtains

(4.30)

With , this leads to

(4.31)

which is the statement of the uncertainty principle. Ehrenfest's theorem

We have noted that observables in quantum mechanics have some uncertainties associated with them. However, wehave also noted that a wave packet simulates classical motion with some uncertainties. If these uncertainties couldbe ignored, the average position would be the same as the position of the centre of a wave packet and we wouldhave classical motion. This is formalised by Ehrenfest's theorem.

Consider the time dependence of the average position,

(4.32)

Similarly, we have for the acceleration,

(4.33)

where is the potential. This suggests that classical equations are implied if we consider average values. In takingaverage values, we are ignoring uncertainties. Therefore, if the uncertainties are small compared with the averagevalues as in the case of macroscopic objects, classical equations of motion are adequate for physical descriptions. Einstein-Padolsky-Rosenfeld paradox

The probabilistic interpretation of measurements in quantum mechanics is very different from a classical point ofview. It leads to many unusual results, one of these is the E-P-R paradox with the collapse of a state. The state ofa system is written as a linear superposition of eigenstates of an operator corresponding to an observable, say ,

(4.34)

An observation of will give one of the values, and after the observation the system will be in the state

and develop according to subsequent Hamiltonian. The collapse to one of the eigenstates leads to some unusualresults. To be specific, consider a particle with zero spin which decays into two particles with equal and opposite

spins, e.g. . Since the total spin is zero, the possible states are

(4.35)

Let the initial state be

(4.36)

After the two particles separate, the system has equal probability to be in either of the two states in Eq.(4.35). Nowif the spin of is measured, it will be found in spin or spin state. But the same measurement will force to

be in states or respectively though no measurement is made on . The result is particularly striking if the

measurement is made after the particles are separated by a large distance. How does the information of themeasurement on the first particle get transmitted almost instantaneously to the location of the second particle? Itmay be noted that any subsequent measurement on the second particle will give a deterministic result.

According to quantum mechanics, the collapse of a state is not a local, causal effect. It is an overall global, non-local change. Quantum mechanics is based on a classical measuring system, and a quantum non-local, global, non-causal system. The classical observation leads to global, non-causal quantum state.

4.3 Step potential

As a first example of a particle in the presence of a potential, we consider a step potential

(4.37

where is the Heaviside step function. The Schroedinger equation is

(4.38)

An important point to note is that when the operators in a linear differential equation depend on separate variables,one can obtain special, separable solutions. Furthermore, the general solution can be obtained as superpositions ofseparable solutions. Taking

(4.39)

and dividing Eq.(4.38) by leads to

(4.40)

In writing the solutions to the spatial equation, we consider specific boundary conditions. For example, forthe beam of particles coming in from the left, we take

(4.41)

The corresponding probability currents are

(4.42)

Here, for the beam is moving to the right for , and for the is taken to be pure positive

imaginary. Now for the second order differential equations, the wave function and the first derivative have to becontinuous which implies that

(4.43)

For the wave function in (4.41), this leads to

(4.44)

which gives

(4.45)

The corresponding reflection coefficient is

(4.46)

and the transmission coefficient is

(4.47)

It may be noted that

(4.48)

which is a consequence of the conservation of probability. For two special cases, one has

(4.49)

Some interesting properties are the following: 1. As in the case of the free particle, the wave function is not normalizable. However one can use it to describe a

beam of incoming particles with density per unit length moving to the right, fraction reflected and

fraction transmitted with .

2. For one takes which implies that with all particles reflected. However the wave

function is non-zero for though it decreases exponentially. Thus the particle can be found in the region

. This is known as barrier penetration.

3. A particularly interesting case is that of . In this case so that

(4.50)

and the corresponding wave function reduces to

(4.51)

Effectively, the solution could have been obtained by requiring that

(4.52)

4. There is reflection even when is negative. Finally, we note that the 1-dimensional step function potential maysimulate a situation when an electron moves in a direction perpendicular to a surface with probability for reflectionand transmission.

4.4 Particle in well, finite/infinite

(4.53)

When is negative, the solutions for positive energies can be considered for beams coming in and reflected or

transmitted, and have to be analysed with continuity conditions at and . But now, there can

be negative energy bound states for . In this case, since the potential and the Hamiltonian are invariant

under parity inversion, one has with being the parity inversion operator, and one can take the

energy eigenstates to be odd or even functions of . Therefore, for negative energy eigenstates, we take

(4.54)

with for even states and for odd states. Continuity of the wave functions and their derivatives at

lead to

(4.55)

which is a non-holonomic equation. Two special cases are interesting. For , with one gets

(4.56)

Neglecting compared with , this leads to

(4.57)

This is an interesting result which is valid to leading order, for all weak, attractive potentials. For , but

finite , i.e. low-lying states, we get

(4.58)

where for even states, and for odd states. These are the energy eigenvalues for a

particle in a box. For the general case we have

(4.59)

Here the function on the left hand side starts at at , decreases gradually and becomes at

which corresponds to . The function on the right hand side is at for , increases

gradually and tends to at . This is repeated between and . Similarly,

for , it starts from at and tends to at . Therefore effectively,

the two sides of Eq.(4.59) intersect times if

(4.60)

and there are bound states.

For the special case of , we can consider an equivalent description,

(4.61)

Then the solutions can be obtained from the condition in Eq.(4.52) that the wave function vanishes at the points atwhich the potential tends to . The solutions are

(4.62)

which vanishes at . For the wave function to vanish at , one should have

(4.63)

(4.67)

where the factor of 2 is compensated by allowing to take positive and negative values. This relation for

the density of states is an essential part of statistical mechanics. 5. The energy eigenstates illustrate Bohr's correspondence principle which states that a quantum systemtends to a classical analogue for large quantum numbers.

For the large energy eigenstate of a particle in a box, the probability of finding the particle between

and is

(4.68)

which is equal to the expected classical probability.

4.5 -function potential

(4.69)

For scattering, one can take for the beam coming in from the left,

(4.70)

Integration of Eq.(4.69)from to across the origin leads to the boundary conditions

(4.71)

From these relations one obtains

(4.72)

for the transmission and reflection coefficients. For the bound states, one has

(4.73)

This implies that we have only one bound state, for . We can also obtain the energy eigenstates directly.

It may be noted that the negative parity states are not affected by the -function at , and we consider onlythe positive parity states. For this we take

(4.74)

Integrating Eq.(4.69) from to leads to

(4.75)

The corresponding normalised wave function is

(4.76)

for which the first derivative is a step function and the second derivative leads to the -function.

4.6 Simple harmonic oscillator

The S.H.O. is often used as a first approximation to potentials with a minimum,

(4.77)

We consider the energy eigenstates,

(4.78)

Since the potential is invariant under parity operation ,

we have energy eigenfunctions with well-defined parity. For obtaining the solutions to the equation, one notes that

(4.79)

where the exponent is chosen for the wave function to be normalizable.

Separating the exponential part, we consider a solution of the form

(4.80)

for which one has for the derivatives,

(4.81)

Substitution of these in Eq.(4.78) multiplied by leads to

(4.82)

Taking

(4.83)

(4.84)

From this, one has for the coefficient of ,

(4.85)

Since the denominator can be written as

(4.86)

one can take

(4.87)

For the confluent hypergeometric equation

(4.88)

one has solutions in the form

(4.89)

for which the ratio of successive terms is

(4.90)

Comparing the relation for in Eq.(4.87) with this, and noting that the successive terms in Eq.(4.83)

have an extra factor of , we deduce

(4.91)

This leads to

(4.92)

where we have included the gaussian term in Eq.(4.80) and the

extra factor of for in Eq.(4.83).

Now the asymptotic behaviour of implies

(4.93)

which implies that . Therefore normalizability of require

that the series for the confluent hypergeometric function in Eq.(4.92) should terminate.

This takes place if

(4.94)

One can take

(4.95)

where for even states with , and for odd states with .

It is interesting to note that the solutions for in Eq.(4.82)

for these energy eigenvalues in Eq.(4.95) are related to Hermite polynomials. Hermite polynomials: For the energies in Eq.(4.95),

he equation for in Eq.(4.82) takes the form

(4.96)

Taking , one gets

(4.97)

The solutions to this are Hermite polynomials

(4.98)

To show this, we note that the polynomials have some recursion relations,

(4.99)

We use these relations to obtain

(4.100)

Substituting these in Eq.(4.97) for , one gets

(4.101)

where we have used the relation in Eq.(4.99). Therefore, the solution

for the eigenfunctions can be written as

(4.102)

It may also be noted that Hermite polynomials have a generating function,

(4.103)

Expansion in powers of leads to

(4.104)

One can then use the recursion relations in Eq.(4.99)

and Eq.(4.100) to obtain the higher order polynomials and their derivatives.

For obtaining the normalization constants in Eq.(4.102), we have

(4.105)

For , one has

(4.106)

There are some interesting and useful relations, scaling, virial relation,

and Feynman-Hellmann theorem, which provide useful physical insight.

Scaling properties: Multiplying Eq.(4,78) by one gets

(4.107)

With scale transformation , this leads to

(4.108)

Taking to make the coefficient of the second term dimensionless, one gets

(4.109)

Comparing this with the original equation in Eq.(4.78) with , one gets

(4.110)

These relations are consistent with the results in Eqs.(4.94) and (4.95).

Virial relation : One can deduce some relations for someexpectation values by using virial relation. Starting with the relation

(4.111)

for energy eigenstates, one gets

(4.112)

For the special case of , this leads to

(4.113)

For the case of the simple harmonic oscillator, , one gets

(4.114)

Feynman-Hellmann Theorem: We will deduce the implications of

Feynman-Hellmann theorem where one considers the dependence on some parameter ,

(4.115)

for energy eigenstates. Taking derivatives with respect to , one gets

(4.116)

Using the property that are eigenstates of , this leads to

(4.117)

For the harmonic oscillator, we take and , which leads to

(4.118)

in agreement with the relation in Eq.(4.114) deduced from virial relation.

4.7 Bound state in a weak potential

Consider a 1-dimensional potential which tends

to zero for . For the energy eigenfunctions we have

(4.119)

For large values of , since , we have

(4.120)

For tending to small values, both and are small, and therefore

both and are small. This implies that does not

change significantly around the origin, and therefore maintains the form

(4.121)

Integrating Eq.(4.119) across the origin, we get

(4.122)

Squaring the two sides, one obtains for the energy,

(4.123)

for and tending to small values, and for the case where the integral is bounded and tends to

zero for . This implies that there is one bound state for attractive 1-d weak potentials, with the energy ineq.(4.123). It may be noted that this result is consistent with the bound state solutions for a particle in a finite well,Eq.(4.57), and for the -function potential in Eq.(4,75),

(4.124)