The Harmonic Oscillator
One of the first tests to which we subject a new theory requires that we apply it to simple problems, usually ones that we have already solved, to see whether it gives us a good, correct solution. In the spirit of that test we take the simple harmonic oscillator as the subject of such a test of the quantum theory, just as the discoverers of the theory did in the mid-1920's. In that simple system we find a small body of mass m attached to one end of a spring of force constant k with the other end of the spring attached to a suitably massive body that effectively does not move when the spring exerts a force upon it. We constrain the system to move in only one direction, conventionally along part of the x-axis of our coordinate system, and note that if we give the small body a certain amount of kinetic energy, it will move along a line segment in a manner that corresponds to a sinusoidal function of elapsed time.
For the force exerted by a spring we have F=-kx, in which equation x represents the distance we pull or push the free end of the spring away from the neutral point, the point that the free end occupies when the spring exerts no force at all. We imagine stretching or compressing the spring and we integrate that force law to calculate the potential energy U that we thus store in the system,
(Eq’n 1)
If we represent the total energy that we put into the system, either through an impact or by stretching/compressing the spring then letting it go free, by E, then we calculate the kinetic energy of the small body at any instant as T=E-U. Substituting the appropriate algebraic descriptions of the body’s kinetic energy and potential energy, we can calculate the body’s velocity at any instant as
(Eq’n 2)
Note that I have tacitly assumed that the spring has negligible mass relative to the mass of the small body, so all of the system’s kinetic energy goes into the body while all of the system’s potential energy goes into the spring. Defining an alpha term as
α=2E/k, we can rearrange that equation into
(Eq’n 3)
We then apply the process of integration to get
(Eq’n 4)
in which we define the angular frequency of the system by writing
(Eq’n 5)
Taking the antilogarithm of Equation 4 gives us
(Eq’n 6)
in which the constant of integration C has become the multiplicative factor A corresponding to the amplitude of the oscillation represented by the imaginary time exponential. Because x2 will always have a value smaller that
α, the second term on the left side of Equation 6 always gives us an imaginary number and the two terms taken together represent the motion of a point on a circle in the complex plane in accordance with the imaginary exponential on the right side of the equality sign. We want only the real values for the location of the small body, so we drop the imaginary terms and Equation 6 becomes
(Eq’n 7)
Now for the quantum mechanical version. In quantum mechanics we don’t calculate a description of the location of the small body directly, as above. Nor, for that matter, can we calculate a description of the body’s velocity, as we did above. Instead we equate the total energy in the system to the Hamiltonian function appropriate to the system, express the two functions in terms of the appropriate mathematical operators, then apply that operator equation (Schrödinger’s Equation) to a putative state function. Solving that second-order partial differential equation gives us the full algebraic form of the state function, which encodes, but does not describe, the dynamic properties of the system. In this case the Schrödinger equation takes the form
(Eq’n 8)
in which ∂x= ∂/∂x.
One of the nicer consequences of the quantum theory’s encoding, rather than describing, the dynamic variables of a system comes to us from the fact that we do not have to solve Schrödinger’s Equation in order to extract information about the system. We don’t actually need an explicit algebraic representation of the state function. Equation 8 looks very much like a quadratic equation, but we can’t solve it as a quadratic equation because what looks like the square of the unknown actually represents a differential operator. Nonetheless, let’s try factoring the Hamiltonian operator and see what we get:
(Eq’n 9)
in which we have the commutator function,
(Eq’n 10)
For convenience we recognize the square root of k/m as the angular frequency of the oscillator and we define two new operators,
(Eq’n 11)
and its adjoint (Hermitian conjugate to mathematicians)
(Eq’n 12)
Thus we have the harmonic oscillator Hamiltonian (Equation 9) as
(Eq’n 13)
Alternatively, we could have written the factors in Equation 9 in the other order, with the minus sign in the second factor, and gotten our Hamiltonian as
(Eq’n 14)
Note that because we often use differential operators in quantum theory, our multiplications, or what look like multiplications, rarely obey the commutative rule. Thus, we need to use commutators.
Next we substitute from either Equation 13 or Equation 14 into Equation 8. Choosing the Hamiltonian of Equation 13, we get
(Eq’n 15)
We calculate the commutator relation in that equation as
(Eq’n 16)
so Equation 15 becomes
(Eq’n 17)
Using the Hamiltonian of Equation 13 gives us in addition
(Eq’n 18)
Subtracting Equation 18 from Equation 17 gives us the commutator relation
(Eq’n 19)
We use that result to work out the commutators of the a-operators with the Hamiltonian operator; thus, we calculate
(Eq’n 20)
In that calculation I exploited the fact that [a*,
ħω/2]=0 because ħω/2 represents a constant and a constant commutes with anything, thereby making the commutator zero out. By the same calculation applied to the other a-operator we also get
(Eq’n 21)
So far we have done no actual physics. We have merely used mathematical manipulations to describe some operators that we will find convenient to use. We often need to take such side excursions into mathematics because a lot of the mathematics that we use in physics lies on some of the less familiar provinces of the Map of Mathematics. It helps to take a little time to discover new tools and to get familiar with them before we use them, so we need to acquaint ourselves with a little mathematics before we get into our real work.
Now we take Schrödinger’s Equation in the form H
ψ=Eψ and apply the a*-operator to it. As I noted above, the state function ψ encodes the dynamic properties of the system, so we use mathematical operators to tease a description of those dynamic properties out of the state function. In this case we don’t even need to know the actual algebraic form of the state function in order to derive a description of the energy that the system can contain. We have
(Eq’n 22)
in which we have used the commutator to reverse the order of a* and H. Note that a* and E commute with each other because E represents a constant, a fixed number. Making the appropriate substitutions from Equation 20 and rearranging the formulae gives us
(Eq’n 23)
The time-independent Schrödinger equation, especially the version displayed as Equation 23, gives us an example of an eigenvalue equation (from German eigen, meaning characteristic or typical of). Equation 23 also tells us that applying the Hamiltonian operator to the state function a*
ψ (equivalent to measuring the energy contained in the state that eigenfunction encodes) gives us an eigenvalue of the energy larger than the eigen-energy in the state encoded in ø by the amount ħω. Applying the a*-operator repeatedly leads us to the statement that
(Eq’n 24)
in which n=0, 1, 2, 3, 4, .... In the same way application of the a-operator to the state function in Schrödinger’s Equation tells us that
(Eq’n 25)
We see that applying the a*- or a-operator to the eigenfunction
ø transforms it into another eigenfunction, one that contains an amount of energy equal to ù more or less than the energy contained in the original eigenfunction. Thus, we refer to those operators as ladder operators and, more specifically, we call the a*-operator a raising operator (or creation operator) and the a-operator a lowering operator (or annihilation operator). Applying those operators to the harmonic oscillator eigenfunction corresponds to the oscillator absorbing or emitting a quantum of energy (presumable by means of a photon or a phonon), so we have effectively derived the assumption that Max Planck had to make in devising his theory of blackbody radiation. And we have done it without knowing the actual algebraic form of the state function.We can add one more fact to what we know about the quantum-mechanical harmonic oscillator. We know that the oscillator cannot go into a state that has negative energy, so we can infer the existence of a minimum-energy state
ψ0 for which
(Eq’n 26)
stands true to Reality. Now we want to determine the amount of energy the oscillator contains when it occupies that state. We have, reversing the terms in Schrödinger’s Equation,
(Eq’n 27)
So the energy eigenvalues of any harmonic oscillator come out to
(Eq’n 28)
Finally we want to work out the explicit algebraic form of the harmonic oscillator state function. For convenience we define two numbers,
(Eq’n 29)
and
(Eq’n 30)
so that we can rewrite our raising and lowering operators as
(Eq’n 31)
and
(Eq’n 32)
Now we solve Equation 26 explicitly. We have
(Eq’n 33)
which gives us
(Eq’n 34)
We multiply that equation by dx, integrate it, and take the antilogarithm of the indefinite integral, thereby getting
(Eq’n 35)
To calculate the state function of the higher energy states we use the procedure implied in Equation 24 and apply the raising operator as many times as necessary, thereby getting
(Eq’n 36)
If we define a new constant by writing
(Eq’n 37)
and divide the operator by the square root of the product of
α and β (αβ= ħω/2), then we can rewrite Equation 36 as
(Eq’n 38)
We can certainly use that equation to calculate the state functions of an harmonic oscillator, but we also notice that it coincides with the operator part of Rodrigues’ formula for the generation of the set of the Hermite polynomials,
(Eq’n 39)
Thus we have
(Eq’n 40)
Now we need to normalize the state function; in other words, we need to ensure that the function satisfies the statement that
(Eq’n 41)
when we carry out the integration over the entire space available to the oscillator. Using the generating relation for Hermite polynomials,
(Eq’n 42)
we can write
(Eq’n 43)
in which we have used the fact that
(Eq’n 44)
Equating the coefficients of the same powers of st, we get
(Eq’n 45)
Comparing that result to Equation 41 necessitates that we rewrite the state function of the n-th state of the harmonic oscillator as
(Eq’n 46)
That function solves the Hermite equation in the form
(Eq’n 47)
in which
(Eq’n 48)
From that latter equation we get
(Eq’n 49)
which we use to justify absorbing the coefficients of Equation 38 into the function in Equation 46.
Thus we have the theory describing the dynamics of an harmonic oscillator in two different versions. In the classical version we tracked the motion of the oscillating body and in the quantum version we tracked the possible energies that the system could possess. We can judge this division as appropriate quite simply. We use classical-level oscillators in mechanical devices, such as clocks, so we need to track the actual motion of the oscillating body. In the quantum-level oscillator, on the other hand, Heisenberg’s indeterminacy principle tells us that we cannot track actual motions of the oscillating body; we can only devise a description of the probability of finding it in a certain location. But we don’t need location information from a quantum oscillator: instead, we calculate allowed energy levels, whose descriptions we use in thermodynamic calculations, such as we find in Planck’s theory of blackbody radiation or in the theory of the specific heats of solid matter.
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