3. Simple Harmonic Oscillator
1
6. Time-Independent (Stationary State) Perturbation Theory
NOTES:
A. Introduction
So far, we have been looking at systems that we can solve completely analytically. The potential energies and Hamiltonians have been of a form that makes the Schroedinger equation solvable without any approximation techniques. What if we don’t have such a “nice” Hamiltonian. In general, one must rely on numerical methods to solve the Schroedinger equation (a differential equation). However, if the Hamiltonian is not too different from one of the nice, solvable Hamiltonians, then we can use the results of perturbation theory to find approximate energies and eigenfunctions of the system. This theory is what we will develop here.
Perturbation theory works best when the system’s total energy is “perturbed” by a small additional potential energy. The total Hamiltonian can be expressed as a sum of an unperturbed Hamiltonian() and a perturbation ():
(6.1)
The unperturbed Hamiltonian is one of the nice, solvable Hamiltonians while the perturbation is a relatively small adjustment to the total energy.
The plan is to expand the system Hamiltonian, and hence the system energy, into a sum of terms involving higher and higher powers of , similar to a Taylor series expansion of a function around some value. Since the perturbation is small, we only have to keep the first few terms (typically two or three terms) to obtain a close approximation to the actual energy.
Because of the analogy to a Taylor series, it is useful to review a Taylor expansion for some function f(x). Let’s take the function
The first three terms of the Taylor series expansion about x=0 are
We view x as the perturbation to the unperturbed value of f(x) which is one. If x is very small compared to one, then the two terms added to one in the expansion are adequate to obtain the value of f(x). The first term (involving x) added to one is the first order correction and the second term (involving x2) is the second order correction. As an example, let x = 0.2. The value of f(x = 0.2) = 1.0954451. Using the first and second order correction, we obtain f(0.2) = 1 + 0.1 - 0.005 = 1.095. This approximate value is only 0.04% less than the actual value.
We will attempt to do a similar expansion to find the corrections to the unperturbed energy to obtain an approximate but very accurate value for the energy of the perturbed system.
B. Nondegenerate State Perturbation
The first procedure that we obtain will only work if a system is in a perturbed state that is very close to a nondegenerate, unperturbed eigenstate of the unperturbed Hamiltonian. This probably won’t be obvious at first, but we will see later why this is so. We will discuss what to do if the unperturbed state is degenerate in a later section.
Let the unperturbed eigenstate be denoted by . The superscript of (0) means that this state is the unperturbed eigenstate of . Let’s let the energy eigenvalue of this state be so that
(6.2)
The subscript n refers to this one particular eigenstate of . Now for some time-saving notation. Let’s just refer to the eigenstate by its label n; we drop the so that the eigenvalue equation (6.2) looks like
(6.3)
We assume that the actual eigenstate of the total Hamiltonian is very close to the unperturbed state and that it can be written as a series expansion of the form
(6.4)
where is the first order correction and the second order correction to . Likewise, we assume that the actual energy E is very close to the unperturbed energy and can be written as a series expansion of the form
(6.5)
where is the first order correction and the second order correction to . The question is how to find these corrections, particularly the energy corrections since finding the energy is usually of primary importance.
To assist us in this endeavor, recall that what gives rise to these series expansions is the perturbation of the Hamiltonian. If the perturbation is “turned off”, then the correction terms vanish; if the perturbation is “turned on”, then the correction terms appear. Let’s introduce an integer variable q such that we write
(6.6)
This q can have two values: if q = 0 then the perturbation is turned off, and if q = 1 then the perturbation is turned on. Using this q, we can rewrite (6.4) and (6.6) as
(6.7)
and (6.8)
Notice, again, that if q = 0 the perturbation is off and we get the unperturbed eigenstate and energy, but if q = 1 the perturbation is on and we get the corrections to the eigenstate and energy. Notice also that the second order corrections involve the square of q. This is logical since we expect a second order term involving the square of to give rise to the second order corrections to the eigenstate and energy.
Now the Schroedinger equation for the system is
(6.9)
Substituting (6.6-6.8) into this equation gives
(6.10)
Now we want any first order corrections to only depend on terms involving , and second order corrections to only depend on terms involving , etc. This is accomplished if we simply require that the sum of terms involving q to a certain power on the left side of (6.10) equal the sum of terms involving q to that same power on the right side of the equation. This is sometimes referred to as “equating the powers” of q. Let’s do this for the first three powers of q.
Equating powers of q0 gives:
(6.11)
This is simply the Schroedinger equation for the unperturbed Hamiltonian that we wrote in (6.2)
Equating powers of q1 gives:
(6.12)
Equating powers of q2 gives:
(6.13)
We will stop here as a second order correction is sufficient for most systems of interest.
First Order Correction to Energy
We take the scalar product of the four vectors in (6.12) with the unperturbed eigenstate. Using Dirac notation, this looks like
(6.14)
Let’s look at the first scalar product in this equation in more detail. Recall that the adjoint of the Hamiltonian is equal to the Hamiltonian since it is an operator corresponding to a dynamic variable (total energy). So we can replace with to get
(6.15)
By definition of the adjoint (refer to (2.21)), the right side of this equation can be written as
(6.16)
Now the energy is real since we can measure it (or more formally, the eigenvalues of operators corresponding to dynamic variables are always real), so the complex conjugate of the energy is simply the energy. Thus, we have shown that
(6.17)
(Notice that we can think ofas acting on the bra vector to give . This is fine since the Hamiltonian is self-adjoint. Since any operator of a dynamic variable is self-adjoint, you can do the same for any operator inside a bracket.)
The energies inside the brackets of the two terms on the right of (6.14) can be simply taken outside the brackets to yield
(6.18)
Assuming that is normalized, we can solve this equation for the first order correction to the energy:
(6.19)
Second Order Correction to Energy
Since we found a successful method to obtain let’s try the same method to find . We take the scalar product of the five vectors in the second order equation (6.13) with the unperturbed eigenstate:
(6.20)
Evaluating these five scalar products gives
(6.21)
and solving for gives
(6.22)
Let’s simplify things by requiring that the corrections to the unperturbed eigenstate be orthogonal to the eigenstate. Thus and we have
(6.23)
Hmm...notice that we need the first order correction to the eigenstate to find the second order correction to the energy. How can we find ?
Look again at the first order equation (6.12). We can rearrange it to yield
(6.24)
This is a differential equation for the function a. We could simply try to solve it for each particular system. A more general approach would be to use as series solution to solve the equation. You can always use a series solution for a differential equation. So let’s try this series approach. We assume that the function a can be expanded as a sum of coefficients times the eigenfunctions of the unperturbed Hamiltonian. This is valid since these eigenfunctions form a complete set of basis vectors. Any vector in the same vector space should be able to formed by the appropriate superposition of these basis vectors. Thus, we write function a as
(6.25)
Some notes about this expression:
1) These eigenfunctions are the eigenfunctions of the unperturbed Hamiltonian . Remember that the eigenfunction of the total Hamiltonian,, is very close in form to one of these unperturbed eigenfunctions which we called .
2) We do not include this particular eigenfunction in the expansion () since we have required that be orthogonal to . Thus, it does not contribute to the formation of .
3) The cm are the coefficients of the expansion. Recall in Chapter 2 that we required these coefficients to be the scalar products between the vector we are forming and the basis vectors, so
(6.26)
Thus, we can write the expansion for the function a as
(6.27)
Now let’s take the scalar product of the vectors in (6.24) with one of the unperturbed eigenstates:
(6.28)
(6.29)
The last scalar product on the right side of this equation is zero since the unperturbed eigenstates are orthogonal. We can then solve this equation for the coefficient cm:
(6.30)
Okay, we’re now ready to combine our results. Let’s substitute (6.30) into (6.27) to get
(6.31)
and let’s substitute this into (6.23) to get an expression for :
(6.32)
Here is the second order correction to the energy. Notice that it does involve the square of the perturbation as we assumed! The energy correction is a sum of potentially an infinite number of terms. How many terms do we need? In practice, you take only a few terms to get an approximate value of . This is ironic since itself is one of the few terms you use to get an approximate value of the system energy!
We need to make a very important point about this expression. Notice that cannot equal or will blow up. Now we already require that which avoids one potential blow up. But if one of the other eigenstates is degenerate with the eigenstate , then and a blow up will occur. Thus, this procedure for finding the second order energy correction is only valid for a perturbed state that closely resembles a nondegenerate, unperturbed eigenstate. That is, cannot be a degenerate state.
C. Almost Harmonic Oscillator
We know that the potential energy of a simple harmonic oscillator is a parabola. Suppose that the potential energy of a system is almost parabolic but with a slight perturbation. Suppose that the potential energy is of the form
(6.33)
where the fourth order term is very small compared to the second order term. We can use the procedure outlined above to find the first order correction to the energy for such an oscillator. Recall that all of the states of the unperturbed oscillator are nondegenerate so that we could use the procedure for any of the states. But let’s focus on the ground state.
The Hamiltonian is where the perturbation is
(6.34)
We assume that the actual state closely resembles the unperturbed ground state, i.e.
(6.35)
and the unperturbed ground state energy is
(6.36)
The first order correction to the energy can be found using (6.19) (Problem [6.1]). The result is
(6.37)
Notice that the first order correction is proportional to but inversely proportional to the square of the mass. One can go on to show that the second order correction to the energy is zero for this potential energy perturbation. You can also show that for any state, the first order correction to the energy is given by
(6.38)
In Problem [6.2], we take a look at the size of this correction for a model of a diatomic molecule.
D. Stark Effect for Ground State Hydrogen Atom
When a Hydrogen atom is situated in an electric field, there is a shift in its energy. This phenomenon is called the Stark effect. (Recall that a shift in energy also occurs if the atom is placed in a magnetic field. This is called the Zeeman effect.) The potential energy of the electron due to the electric field is typically much smaller than the potential energy due to the Coulomb attraction of the proton. Thus, this potential energy due to the field is the perturbation. If we consider the effect of this perturbation on an atom in the ground state, then we can use our nondegenerate perturbation theory since the ground state of a Hydrogen atom is a nondegenerate state.
The unperturbed Hamiltonian is simply the Hamiltonian of an ordinary Hydrogen atom; it is the sum of the kinetic energy and the Coulomb potential energy:
(6.39)
where k is Coulomb’s constant. The unperturbed eigenstate that closely resembles the perturbed eigenstate is the ground state:
(6.40)
We have written out the function for the ground state since we will have to actually perform integrations in evaluating scalar products. (Sorry..there is no way to avoid this!) Table 7.2 on page 243 in Eisberg & Resnick lists the first few eigenfunctions. (That table is also duplicated in the problem section of this chapter.) Recall that ao is the Bohr radius and equal to 0.53 Å. The unperturbed energy is
(6.41)
What is the form of the perturbation ? This will be the potential energy due to the interaction of the electron with the electric field. Let’s call the electric field . Recall from Physics II that the work done by the electric field in moving a charged particle along some path is given by
(6.42)
Work is also the difference between the potential energy at the initial position and the potential energy at the final position:
(6.43)
Over the small displacement of , the change in potential energy is then
(6.44)
So we conclude that the potential energy at some position is given by
(6.45)
Let’s assume that the electric field is uniform with a magnitude of and a direction along the +z axis. Then the perturbation to the Hamiltonian is
(6.46)
We are now ready to apply our recipes for finding the corrections to the unperturbed energy of the atom as summarized in (6.19) and (6.32). You will do this in Problems [6.3] and [6.4]. Let’s discuss the results of these problems here.
Discussion of Stark Effect in Ground State Hydrogen
First, let’s look at the size of this perturbation to the potential energy. Taking the radius of the ground state atom to be the Bohr radius, the Coulomb potential energy is
(Do you remember that you showed that the potential energy is twice the total energy for a 1/r2 force? This is the “virial theorem”.) Let’s pick a realistic electric field size of 10000 V/m. This could be produced easily in the lab by applying a voltage difference of 100 V across a set of parallel plates separated by 1 cm. With this value for the field, the maximum potential energy of the electron would be
This is approximately 50 million times smaller than the Coulomb potential energy! This is indeed a small perturbation, perhaps so small that the perturbation is negligible? We will see. A quick calculation shows that if the potential energy due to the field is to be comparable to the Coulomb potential energy, then the field has to be on the order of 500 billion Volts per meter (51011 V/m)! Thus, we can say with some confidence that any electric field we can apply to the atom can be treated as a perturbation.
You will show in Problem [6.3] that the first order correction to the ground state energy is zero. This tells us that the perturbation to the energy is going to be small. You will show in Problem [6.4] that the second order correction is
(6.47)
Note that the electric field lowers the energy of the electron since is negative and the unperturbed energy is also negative (-13.6 eV). Also note that the second order correction to the energy depends on the square of the field as we expect (since the Hamiltonian perturbation depends linearly on the field size).
Taking our field value of 10000 V/m from above, the energy is lowered by
This shift is undetectable! What kind of shift is detectable? Recall that the spin-orbit interaction produced an energy shift on the order of 10-4 eV. This is detectable using optical spectroscopy and measuring the transition wavelengths between shifted energy levels. The wavelength of a photon emitted in a transition is given by
where E is the transition energy. Differentiating both sides of this equation will tell us how a perturbation to the energy affects the wavelength:
For example, if the atom makes a transition from the n=2 state (E2 = -3.4 eV) to the ground state (E1 = -13.6 eV) , the transition energy in the absence of any perturbations is 10.2 eV. If the transition energy is lowered by 10-4 eV, then the transition wavelength increases by
This is about the limit of resolution in spectroscopy so that energy shifts smaller than 10-4 eV are just too small to detect.
Suppose we did just want to see the perturbation due to the Stark effect. Solving (6.47) for the electric field necessary to give a second order correction of = -10-4 eV = -1.610-23 J gives
This is an extremely large field! (Air experiences dielectric breakdown at about 3106 V/m.) Wouldn’t this extremely large field exert a tremendous force on the electron and actually ionize the atom (i.e. strip the electron from the proton)? No! This electric field would exert a force of
on the electron. The Coulomb force holding the electron to the proton is
The Coulomb force is approximately 600 times stronger than the electric force, even with this extremely large field. The atom will not ionize!
What can we say in summary about the Stark effect in ground state Hydrogen? The effect is extremely small. For it to be easily detected requires extremely large fields on the order of hundreds of millions of Volts per meter.
A final note of interest should be made between magnetic field effects and electric field effects on the Hydrogen atom. Recall that magnetic field effects are readily observable. The spin-orbit interaction produces a detectable energy splitting with an internal magnetic field produced by the orbiting electron that is approximately 1 Tesla. An external field that is comparable to this or greater can produce larger energy shifts.