Chapter 8

Linear Harmonic Oscillator—Revisited

8.1  INTRODUCTION

One-dimensional harmonic oscillator problem was studied in Chapter 6, where Schrodinger equation was solved using the power series method. This method is a bit lengthy but it has the virtue that same strategy may be applicable to other potentials, for example, Coulomb potential (which we shall discuss in Chapter 10, in the context of Hydrogen atom). The second method of solving Schrodinger equation for harmonic oscillator is the algebraic technique using the so-called annihilation and creation operators or lowering and raising operators or ladder operators. This method is really quicker and simpler too (in comparison to the power series method). This technique shall also be used in connection with the theory of angular momentum. In fact, a generalization of this technique has been applied to a broad class of potentials in supersymmetric quantum mechanics. We start with describing the algebraic technique of annihilation and creation operators in the next section.

8.2  THE CREATION AND ANNIHILATION OPERATORS

The Hamiltonian for a linear harmonic oscillator has the form

equation

where symbols have their usual meaning. The operators equation and equation have commutation relation.

equation

Also, it can be checked easily that the commutators of operators equation and equation with the Hamiltonian operator Ĥ are

equation
equation

Let equation be an eigenstate of Ĥ, with En its (energy) eigenvalue, so

equation

The Hamiltonian equation might be factored like equation if equation and equation were not operators (or if equation and equation were commuting operators). Now when equation and equation are non-commuting operators, we have

equation

We now introduce a special notation for the operator factors appearing on R.H.S. of Eq. (8.6). Let us write

equation
equation

Here operator â is Hermitian conjugate of operator â. In terms of â and â, Ĥ may be written as

equation

We can easily evaluate various commutators,

[â, â] = â â − â â
equation
= 1        (8.9a)

 

[â, â] = [â, â] = 0        (8.9b)

 

[â, â â] = â [â, â â] + [â, â] a = â        (8.9c)

 

, â â] = â, â] + [â, â] â = −â        (8.9d)

 

[â, Ĥ] = ħω [â, â â] = ħωâ        (8.9e)

 

, Ĥ] = ħω, â â] = − ħωâ        (8.9f)

 

Let us now see what happens when operator â operates on eigenket equation. To see this, let us have following operation:

equation
equation
equation

Comparing Eqs (8.5) and (8.10), it is clear that as equation is an eigenstate of Ĥ (with corresponding eigenvalue En), equation is also an eigenstate of Ĥ with eigenvalue (En + ħω). So operator â operating on eigenstate equation changes it to a state with energy (En + ħω), that is, the energy increases by ħω. Therefore, â is called as raising operator (or creation operator). It may be easily checked that equation is also an eigenstate of Ĥ with eigenvalue (Enħω). The operator â, which when operating on a state equation changes it to a state with energy (Enħω) (i.e. the energy decreases by ħω) is called lowering operator or annihilation operator). If operator â operates again on state equation [with eigenvalue (Enħω)], this state is further changed to a state of eigenvalue (En − 2ħω), and so on. Let us suppose we go on operating the operator â successively on the eigenstate, we shall get states with decreasing energies (En − 3ħω), (En − 4ħω), and so on, until we reach the ground state equation. If we now apply operator â on this ground state equation, by the very definition of ground state, the operator is not able to change it to further lower energy state. So â operating on ground state equation should give zero.

Therefore,

equation

or

equation

using Eq. (8.8), we have

equation

or

equation

or

equation

So, if we start with ground state equation, which has energy eigenvalue equation, and operate raising operator â on it, we shall get a state with energy equation. Let us call this state as equation. Again operating â on equation shall produce a state (say equation) of energy eigenvalue equation, and so on. Thus, we get successive eigenstates equation, equation, equation, … with corresponding energies equation, … Thus state equation has energy eigenvalue equation.

So, we get

equation

The raising operator â operating on eigenstate equation (of energy eigenvalue En) changes it to a eigenstate of energy En + ħω; increases the energy of the oscillator by quantum of energy ħω. That is why the operator â may be called as creation operator. By same logic the lowering operator â may be called as annihilation operator.

8.3  ENERGY EIGENSTATES

The annihilation operator â operating on eigenket equation of Ĥ produces a state which is again an eigenket of Ĥ (with eigenvalue En − ħω = En −1). So state equation is nothing but a state equation multiplied by a constant, say λn, that is

equation

So

equation

Taking Hermitian conjugate of Eq. (8.14b) gives

equation

Now

equation

Also, from Eq. (8.8)

equation

Therefore,

equation

so

|λn|2 = n

or

equation

and Eq. (8.14a) gives

equation

Similarly, it can be seen that

equation

or

equation

Using Eq. (8.19), we can easily get

equation
equation
equation
equation

The fact remains that, though, we have found the energy eigenvalues equation, we have still to find the forms of the eigenstates equation, equation, equation, … and so on. Below, we proceed to find these eigenstates.

Let us start with Eq. (8.11)

equation

Here, ground state is represented by the eigenket equation, or by ψ0 (x). Writing explicit form of annihilation operator â from Eq. (8.7a), Eq. (8.11) gives

equation

or

equation

or

equation

Integrating this equation, we get

equation

or

equation

where

equation

We can easily get the value of constant A by normalizing the wave function ψ0(x). We get

equation

This is the same ground state wave function ψ0(x) [given in Eq. (6.49a)] obtained by solving Schrodinger differential equation for harmonic oscillator. We may now find out wave function ψ1(x) using the relation (8.20a)

ψ1(x) = â ψ0(x)
equation
equation

Similarly, ψ2 (x) may be found from ψ1(x) using

equation

which gives

equation

Similarly, we get

equation
equation

where Hn(βx) is the Hermite polynomial.

8.4  MATRIX REPRESENTATION OF VARIOUS OPERATORS

From Eq. (8.18) we have

equation
equation

We may find matrix form of operator â by finding all its matrix elements equation in the basis of energy eigenkets equation.

equation

Similarly,

equation

Thus â and â are represented by the matrices.

equation

and

equation

These are non-Hermitian matrices.

From Eqs. (8.7), we have

equation
equation

So,

equation
equation

These are Hermitian matrices.

From Eq. (8.8), we have

equation

.·.

equation
equation

Therefore,

equation
8.5  EXPECTATION VALUES OF VARIOUS OPERATORS

In Dirac notations, the expectation value of an operator equation in the n-th state of a harmonic oscillator is written as

equation

which, in the usual ψn(x) representation means

equation

Let us evaluate the expectation values of operators equation and equation in eigenket equation.

equation
= zero        (8.31a)

Similarly,

equation
= zero        (8.31b)

Next, let us find equation and equation

equation
equation
equation

It may easily be seen that the uncertainty product (Δx)np)n for harmonic oscillator state equation, using the definitions

equation
equation

comes out to be

equation

For the ground state, Eq. (8.34) gives

equation

Let us evaluate the expectation value of operator equation in eigenket equation.

equation

It can be easily checked that only terms which have equal number of creation and annihilation operators have given the non-zero values.

equation
8.6  THE COHERENT STATES

In previous section, we observed that among the stationary states equation of the harmonic oscillator, only n = 0 state, that is, the ground state equation hits the uncertainty limit ∆x ∆p = (ħ / 2); in general, equation [Eq.(8.39)]. We shall see in this section that certain linear combinations of eigenstates equation (known as coherent states) also minimize the uncertainty product. These states are the eigenstates of the annihilation operator â.

equation

Here eigenvalue α (of â) is, in general, a complex number.

In fact, we start with the eigenvalue equation (8.37) and proceed to solve it. We shall find that states equation, called as coherent states, have many striking features, for example, (i) these are the states with minimum uncertainty product (as mentioned above) and (ii) these states are the quantum mechanical analog of the classical oscillator.

It may be noted that a trivial solution of Eq. (8.37) is the ground state equation with α = 0 as seen from Eq. (8.11). To solve Eq. (8.37), let us firstly put it in the form

equation

or

equation

or

equation

Integrating it we get

equation

or

equation
equation

After normalizing ψα(x), we get

equation

For α = 0, this is the ground state wave function of harmonic oscillator.

Let us go back to Eq. (8.37) and solve it again; this time by expanding eigenket equation in terms of complete eigenket set equation:

Let

equation

so

equation

Now,

equation

so

equation

Comparing coefficient of same eigenket on both sides, we get

equation

which dictates the relations

equation

Putting Eq. (8.45) in Eq. (8.42), we get

equation

Let us normalize equation

equation

so

equation

Therefore,

equation

equation is the normalized coherent state.

Let us look at some of characteristic properties of the coherent states. Firstly, let us see if two different coherent states are orthogonal to each other. We consider two coherent states equation and equation (i.e., two eigenkets of annihilation operator â). Then

equation
equation

or

equation

So, it is clear that different coherent states equation and equation are not orthogonal to each other.

Let us find out expectation value of number operator â â (say equation) in coherent state equation.

equation

Now, equation

So

equation

Similarly, we may get

equation
= |α|4 + |α|2        (8.51)

So the root mean square deviation ∆n is

equation

and

equation
8.7  TIME EVOLUTION OF THE COHERENT STATE AND ITS COMPARISON WITH CLASSICAL OSCILLATOR

If ψn(x) is the solution of time-independent Schrodinger equation, the corresponding time-dependent wave function is written as

equation

If, instead, a general solution of time-independent Schrodinger equation is

equation

then the corresponding time-dependent wave function is written as

equation

We know the time-independent coherent state equation is written as

equation
equation

Therefore, the time evolution of the coherent state may be given by

equation

We can easily check that equation is normalized for all values of time t

equation

Let us now find out the expectation values of operators equation, and equation in state equation. Let us call these time-dependent quantities as equation and equation.

equation
equation

The operators equation and equation may be expressed as linear combinations of creation and annihilation operators â and â [Eqs. (8.26)]. Therefore, to evaluate Eq. (8.58), we need to evaluate the expressions equation and equation.

equation
equation

or

equation

If we take complex conjugate of Eq. (8.59a), we get

equation

So,

equation

If we write α = | α |e, we get

equation

where

equation

Similarly, we get

equation
= − mωxo sin (ωtϕ)
= − po sin (ωtϕ)        (8.61a)

where

equation

From Eqs (8.60a) and (8.61a), it is clear that

equation

This is Newton’s equation of motion, valid for the motion of classical particle. Equations (8.60a) and (8.61a) are the classical equations for the position and linear momentum at time t of a particle in a harmonic potential. Thus, we see that when α is large, the description of a particle (in harmonic potential) through the eigenket equation (or equation) is close to its classical description.

8.8  THE SCHRODINGER AND HEISENBERG PICTURES

We know that the time-development of the state describing quantum mechanical system is obtained by solving the time-dependent Schrodinger equation:

equation

If the Hamiltonian of the system Ĥ is time-independent, Eq. (8.63) may be integrated to give

equation

In fact, what we have stated above comes under the term Schrodinger picture. In Schrodinger picture, the operators equation, equation, Ĥ and so on are time-independent and the time-development of the system is described through the time-dependent state equation. It may be mentioned here that in Eq. (8.64), the exponential of an operator is defined through the power series expansion, that is,

equation

Keeping the fact in mind that the Hamiltonian Ĥ is Hermitian operator, we may easily check that the operator, appearing in Eq. (8.64)

 

Û (t) = eiĤ t/ħ        (8.66)

is a unitary operator. Therefore, the state equation changing to state equation, through the unitary operator Û(t), is a unitary transformation. Equation (8.64) may be written as

equation

where the unitary operator Û (t) may be called as evolution operator. We designate the state equation as equation and operator  as Âs in the Schrodinger picture. The expectation value of an operator  is given as

equation
equation

where the operator ÂH(t) is defined as

ÂH(t) = eiĤt/ħÂ eiĤt/ħ        (8.70)

 

So the expectation value of operator  has been written in two different ways: (i) In Eq. (8.68), the expectation value of  is written with respect to the time-dependent states equation, while the operator  is time-independent.(ii) In Eq. (8.69), the expectation value is written with respect to time-independent states equation while the operator ÂH(t) carries the entire time-dependence. The first one is the (already mentioned) Schrodinger picture, whereas the second one is known as the Heisenberg picture [and hence the subscript H on operator ÂH(t)]. So in the Schrodinger picture, the operators are time-independent and the states are time-dependent, whereas in the Heisenberg picture, the operators are time-dependent and the states are time-independent.

Now taking  to be time-independent, we have from Eq. (8.70):

equation

or

equation

This is called the Heisenberg equation of motion and it gives the time-dependence of an operator.

EXERCISES

Exercise 8.1

Which of the following operators commute with the Hamiltonian Ĥ of one-dimensional harmonic oscillator?

  1. equation
  2. equation
  3. equation
  4. equation
  5. â
  6. â
  7. â2
  8. ↠2
  9. equation
  10. equation

Exercise 8.2

Show that in the nth eigenstate of the harmonic oscillator, the average kinetic energy equation is equal to the average potential energy equation (the virial theorem for harmonic oscillator).

Exercise 8.3

Consider a harmonic oscillator in the superposition state

equation
  1. Find out ψ(x, t).
  2. Show that in state ψ(x, t) the expectation value, equation.

Exercise 8.4

A particle in a harmonic potential is in a state described by the wave function, ψ(x, 0) = A [ψ0(x) + 2 ψn(x)] where ψ0(x) and ψn(x) (n ≠ 0) are normalized eigenstates.

  1. Find the normalization constant A.
  2. Find ψ(x, t) and (x, t)|2.
  3. If one measures energy of this particle, what value shall one get?

Exercise 8.5

Evaluate the following matrix elements.

  1. equation
  2. equation
  3. equation
  4. equation

Exercise 8.6

Evaluate the following matrix elements.

  1. equation
  2. equation
  3. equation
  4. equation

Exercise 8.7

The Hamiltonian of a particle is

 

Ĥ = Aâ â + B + â)

where A and B are constants. Find the energy eigenvalues of the particle.

SOLUTIONS

Solution 8.1

We know the commutator relation

equation

Let us remember the basic commutators

[â â] = 1
[â, â] = 0
, â] = 0
equation

(a)

equation

equation
equation
= [â, â â] ħω
= [â, â] âħω = âħω
, Ĥ] = [â, â â] ħω = − â ħω

equation

Similarly, other commutators may be evaluated. Let us next evaluate equation.

equation
= [â a, â a] ħω
= 0

Similarly, equation

Solution 8.2

The kinetic energy operator equation is

equation

equation

The potential energy equation

equation

So, we find

equation

Solution 8.3

  1. The time-dependent state ψ(x, t) may be written
    equation
  2. equation
    equation

Solution 8.4

  1. For normalization
    equation
    = A2 [1 + 0 + 0 + 4]
    = 5 A2 = 1

    so

    equation
  2.  
    equation
  3.  
    equation

Solution 8.5

  1.  
    equation
  2.  
    equation
  3.  
    equation
  4.  
    equation

Solution 8.6

  1.  
    equation
  2.  
    equation
  3.  
    equation
  4.  
    equation

Solution 8.7

Let us introduce the operator

equation

where S is a constant. Hamiltonian Ĥ becomes

equation

Let us take equation

so

equation

This Hamiltonian gives energy eigenvalues

equation

where n = 0, 1, 2,

REFERENCES
  1. Cohen–Tannoudji, C., Diu, B. and Laloe, F. 1977. Quantum Mechanics. Vol. I and Vol. II, New York: John Wiley.
  2. Griffith D. 1995. Introduction to Quantum Mechanics. New York: Prentice Hall, Englewood Cliffs.
  3. Powell, J. L. and Crasemann, B. 1961. Quantum Mechanics. Reading, Mass: Addison-wesley.
  4. Liboff, R. L. 1992. Introductory Quantum Mechanics. Reading, Mass: Addison-Wesley.
  5. Levi, A. F. J. 2003. Applied Quantum Mechanics. Cambridge: Cambridge University Press.
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