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

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

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

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

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

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

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

We can easily evaluate various commutators,

 

 

 

 

 

 

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

Comparing Eqs (8.5) and (8.10), it is clear that as is an eigenstate of Ĥ (with corresponding eigenvalue E_n), is also an eigenstate of Ĥ with eigenvalue (E_n + ħω). So operator â^† operating on eigenstate changes it to a state with energy (E_n + ħω), that is, the energy increases by ħω. Therefore, â^† is called as raising operator (or creation operator). It may be easily checked that is also an eigenstate of Ĥ with eigenvalue (E_n − ħω). The operator â, which when operating on a state changes it to a state with energy (E_n − ħω) (i.e. the energy decreases by ħω) is called lowering operator or annihilation operator). If operator â operates again on state [with eigenvalue (E_n – ħω)], this state is further changed to a state of eigenvalue (E_n − 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 (E_n − 3ħω), (E_n − 4ħω), and so on, until we reach the ground state . If we now apply operator â on this ground state , 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 should give zero.

Therefore,

or

using Eq. (8.8), we have

or

or

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

So, we get

The raising operator â^† operating on eigenstate (of energy eigenvalue E_n) changes it to a eigenstate of energy E_n + 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 of Ĥ produces a state which is again an eigenket of Ĥ (with eigenvalue E_n − ħω = E_{n −1}). So state is nothing but a state multiplied by a constant, say λ_n, that is

So

Taking Hermitian conjugate of Eq. (8.14b) gives

Now

Also, from Eq. (8.8)

Therefore,

so

or

and Eq. (8.14a) gives

Similarly, it can be seen that

or

Using Eq. (8.19), we can easily get

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

Let us start with Eq. (8.11)

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

or

or

Integrating this equation, we get

or

where

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

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)

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

which gives

Similarly, we get

where H_n(βx) is the Hermite polynomial.

8.4  MATRIX REPRESENTATION OF VARIOUS OPERATORS

From Eq. (8.18) we have

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

Similarly,

Thus â and â^† are represented by the matrices.

and

These are non-Hermitian matrices.

From Eqs. (8.7), we have

So,

These are Hermitian matrices.

From Eq. (8.8), we have

.·.

Therefore,

8.5  EXPECTATION VALUES OF VARIOUS OPERATORS

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

which, in the usual ψ_n(x) representation means

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

Similarly,

Next, let us find and

It may easily be seen that the uncertainty product (Δx)_n (Δp)_n for harmonic oscillator state , using the definitions

comes out to be

For the ground state, Eq. (8.34) gives

Let us evaluate the expectation value of operator in eigenket .

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

8.6  THE COHERENT STATES

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

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 , 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 with α = 0 as seen from Eq. (8.11). To solve Eq. (8.37), let us firstly put it in the form

or

or

Integrating it we get

or

After normalizing ψ_α(x), we get

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 in terms of complete eigenket set :

Let

so

Now,

so

Comparing coefficient of same eigenket on both sides, we get

which dictates the relations

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

Let us normalize

so

Therefore,

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 and (i.e., two eigenkets of annihilation operator â). Then

or

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

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

Now,

So

Similarly, we may get

So the root mean square deviation ∆n is

and

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

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

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

then the corresponding time-dependent wave function is written as

We know the time-independent coherent state is written as

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

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

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

The operators and 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 and .

or

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

So,

If we write α = | α |e^iϕ, we get

where

Similarly, we get

where

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

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 (or ) 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:

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

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

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

 

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

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

where the operator Â_H(t) is defined as

 

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 , while the operator  is time-independent.(ii) In Eq. (8.69), the expectation value is written with respect to time-independent states while the operator Â_H(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 Â_H(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):

or

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. 
2. 
3. 
4. 
5. â
6. â^†
7. â²
8. â^{† 2}
9. 
10. 

Exercise 8.2

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

Exercise 8.3

Consider a harmonic oscillator in the superposition state

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

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 ψ_n(x)] where ψ₀(x) and ψ_n(x) (n ≠ 0) are normalized eigenstates.

1. Find the normalization constant A.
2. Find ψ(x, t) and |ψ(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. 
2. 
3. 
4. 

Exercise 8.6

Evaluate the following matrix elements.

1. 
2. 
3. 
4. 

Exercise 8.7

The Hamiltonian of a particle is

 

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

SOLUTIONS

Solution 8.1

We know the commutator relation

Let us remember the basic commutators

(a)

∴

∴

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

Similarly,

Solution 8.2

The kinetic energy operator is

∴

The potential energy

∴

So, we find

Solution 8.3

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

Solution 8.4

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

   so

   
2.  
   
3.  
   

Solution 8.5

1.  
   
2.  
   
3.  
   
4.  
   

Solution 8.6

1.  
   
2.  
   
3.  
   
4.  
   

Solution 8.7

Let us introduce the operator

where S is a constant. Hamiltonian Ĥ becomes

Let us take

so

This Hamiltonian gives energy eigenvalues

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.