The atomic mass of argon is 39.95 g/mol. What is the natural frequency of two argon atoms bonded together?
First, we determine the mass of one argon atom:
m = 0 03995× = × −
6 022 10. 23 6 63 1026
. kg/mol .
atoms/mol kg/atom
The reduced mass, mr, of a pair of identical argon atoms is therefore
mr
We already calculated the spring constant, k = 0.82 N/m, for an Ar2 molecule in “Back-of-the-Envelope 5.4.” Using this value we can substitute into Equation 5.30:
f k
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Figure 5.12 shows this function plotted over the Lennard–Jones curve. It is exactly the same parabolic curve observed when we previously examined the potential energy of a simple harmonic oscillator, such as a beam (see Figure 5.5). In the region near xe, where vibrations are small, this parabola is a good approximation of the Lennard–Jones potential.
If we set up our axes such that xe (the point of minimum potential energy) occurs at the origin, the graph of potential energy is as shown in Figure 5.13. Since most atomic vibra-tion occurs around this point anyway, the parabolic potential energy curve is an appropri-ate model. Applicable as well, in fact, is the kinetic energy curve observed in Figure 5.5 (not shown again here).
However, there is an important detail that separates these models from quantum mechanics models—it is that these curves are continuous. And since the total energy of an oscillating system is the sum of both potential energy and the kinetic energy, the total energy is also continuous over all values of x. However, we know from quantum mechanics that the energy of atoms can only take on discrete values.
So what values of energy are allowed in a “quantum harmonic oscillator?” The answer can be determined using the Uncertainty Principle discussed in Chapter 3, “Introduction to Nanoscale Physics.”
We start with the total energy, E, of the system. This is the sum of the kinetic and poten-tial energies:
E KE PE= + =1m v + kx
2 1
2 2 2
r (5.31)
Here, mr is the reduced mass of the two-atom system; v is the velocity, defined as the change in the separation distance, x, during a given change in time, t (v = Δx/Δt); and k is the
PE
Lennard–Jones
x kx2 21
xe = 0
FIGURE 5.12 Approximating the Lennard–Jones potential energy with a parabolic curve. The function PE = kx2/2, where k is the spring constant and x is the displacement from the equilibrium separation, matches the Lennard–Jones curve especially well near the equilibrium separation point, xe, as shown in the inset.
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spring constant. (Note that we have simplified the equation for PE by putting xe at the origin, so xe = 0.)
The Uncertainty Principle states that the uncertainty of an object’s position, Δx, and the uncertainty of its momentum, Δp, at any moment in time must be such that
Δ Δx p h
x ≥ 4π
Here, h is Planck’s constant and h = 6.626 × 10−34 J s. So, according to this principle, the energy of the atoms in the quantum harmonic oscillator model can be no less than a fixed amount. Experimentally, if we knew the position of the atoms enough to say they were located at exactly the equilibrium separation, the velocity of the atoms (their change in separation in a given time) would be completely undetermined; if instead, we knew the velocity to be exactly zero, then the atoms’ separation could be any value. Thus to describe a realistic scenario, we are forced to compromise by saying that the lowest energy allowed is where neither the speed nor the position are exactly zero.
Recall that momentum, p = mv, and natural frequency, ωn = (k/m)1/2, so k = ωn2m. This enables us to substitute into Equation 5.31:
E m pm m x
FIGURE 5.13 The potential energy PE of a simple harmonic oscillator as a function of separation distance, x. We have assigned the equilibrium separation as xe = 0.
Nanomechanics ◾ 147 The minimum energy is determined by the smallest allowable values for p and x—that is, their uncertainties—so we replace x with Δx and p with Δp = h/(4πΔx) so that
E h
We are looking for the lowest possible energy and this can be found by taking the deriv-ative with respect to position and then setting it equal to zero (corresponding to the bottom of the parabola, where the slope is zero):
ddE r n r
When we plug this uncertainty in position back into the energy equation we find the mini-mum allowed energy, E0:
This can also be expressed using fn, measured in Hertz (Hz):
E hf
0= 2n (5.34)
So that it is, the smallest amount of energy a quantum harmonic oscillator can have. What this suggests of course is that the system cannot have zero energy, even at absolute zero temperature. This is a significant mathematical result. The ground state of a quantum har-monic oscillator is known as the “zero-point energy.” The energy values are discrete there-after, proceeding upward according to the quantum number n, and the allowable quantum energy, hf (see Section 3.3.3), based on Planck’s quantum hypothesis:
E=⎛n+ hf n
⎝⎜
⎞
⎠⎟ =
12 n 0, , , ,… 1 2 3 (5.35)
Here, h is Planck’s constant and fn is the natural frequency, expressed in Hertz.
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We can see these allowed values of energy on the graph in Figure 5.14. While the classi-cal potential energy curve drops all the way to zero at the equilibrium separation, xe, the lowest allowed energy for the quantum harmonic oscillator is actually E0 = hfn/2. From this zero point, the quantized energies increase by the quantum number, as shown.
5.3.2.3 The Schrödinger Equation and the Correspondence Principle
At the dawn of the twentieth century, experiments showed that subatomic particles could have wave-like properties. Electrons, for example, created diffraction patterns when sent through a double slit. So did photons. Scientists scrambled to find an equa-tion that would describe the behavior of atomic particles. This was discussed in Chapter 3,
“Introduction to Nanoscale Physics.” In that chapter, we learned that a particle of mat-ter can be described by a generalized wave function, denoted by ψ (“psi”). All the
mea-E kx2
PE =21
xe (xe= 0)
hfn E0=
E1= E0+1hfn E2= E0+2hfn
E3= E0+3hfn
21 hfn
x x
x
hfn)
E = (n+ 21 n = 0, 1, 2, 3, ...
FIGURE 5.14 The quantum harmonic oscillator. The classical potential energy curve drops to zero at the equilibrium separation x = xe. However, it was determined that the lowest allowed energy for the quantum harmonic oscillator is E0 = hfn/2, where h is Planck’s constant and fn is the natural frequency (in Hertz).
Nanomechanics ◾ 149 surable quantities of a particle—including its energy and momentum—can be determined using ψ. And the absolute value of the square of the wave function, |ψ|2, is proportional to the probability that the particle occupies a given space at a given time.
When we confine a particle inside a “box” of infinite height, the wave function, ψ is exactly the same as for the vertical displacement of a string stretched horizontally between two walls.
The wave function, ψ, satisfies an important equation that describes the behavior of atomic particles. The equation was developed by Erwin Schrödinger (1887–1961), an Austrian theoretical physicist widely regarded as the creator of wave mechanics. The now-famous Schrödinger equation applies to any confined particle and for motion in one dimension along the x-axis. It is written as
dd
Here, m is the particle’s mass, h is Planck’s constant, E is the total energy of the system, and PE is the potential energy of the system. This is known as the “time-independent”
Schrödinger equation because it is invariant with time. While solving it can be quite com-plicated, this equation has proven extremely accurate in explaining the behavior of atomic systems. And it is used to find the allowed energy levels of quantum mechanical systems, including atoms.
The quantum harmonic oscillator system is exactly the kind of system the Schrödinger equation can be used for. We have a pair of atoms, the total mass of which is combined and treated as a single particle with a reduced mass, mr. The motion of this system is confined to the x-axis, where displacement from the equilibrium position is given by x.
Combining what we know from Equations 5.16 and 5.30, we can find the potential energy of this system to be PE=2π2 2fnmrx , where f2 n is the natural frequency of the system (Hz). Substituting this into the Schrödinger equation, we get
dd r n
This equation requires a lengthy solution, which we do not show here. In the end, we find a solution that contains wave functions at the corresponding allowed energy levels:
E=⎛n+ hf n Uncertainty Principle as our guide. In addition, the solution of the Schrödinger equation in this case provides the wave functions, ψ, and probability distributions, |ψ|2, for the quantum states of the oscillator. We have graphed solutions in Figure 5.15 for the first three
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quantum states, which correspond to the lowest three energies (n = 0, 1, 2). Here we see that the probability of finding the oscillator with a given separation, x, actually drops to zero in certain locations between the oscillator’s maximum amplitudes. This of course is completely contrary to classical mechanics, which predicts a continuum of allowed separa-tions where x can be any value between the maximum separation, A, and minimum separation, −A.
Where else do the classic and quantum descriptions diverge? Well, recall the classical probability distribution, p(x), which predicted the odds of finding an oscillating mass at a given value of x. This curve is shown again in Figure 5.16. As expected, the probability of finding the mass is smallest near the equilibrium point, where it is always moving the quickest and therefore spending the least time. The probability rises fast near the limits of motion (A and −A), where the mass lingers while turning around.
Compare this to the probability density for the lowest energy state of a quantum har-monic oscillator, |ψ0|2, also shown in Figure 5.16. It is almost the complete opposite of the classical mechanics case. Here, the chance of finding the oscillating atom is highest near the equilibrium point. However, this disagreement becomes less and less striking at higher energies. Figure 5.16 shows |ψ2|2, with its three humps of probability. Already we see that the probability near the equilibrium point is diminishing while the probability near the
x
FIGURE 5.15 The wave functions, ψ, and probability distributions, |ψ|2, of the quantum harmonic oscillator. These are determined by solving the Schrödinger equation. The probability of finding the oscillating mass (atom) at a given value of x is given by |ψ|2. Shown here are the lowest three energies, where the quantum number n equals 0, 1, and 2.
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edges increases. We can see why this happens if we look again at the time-independent Schrödinger equation:
From this equation, it is evident why the wave function, ψ, will oscillate less rapidly at the edges than it does near the equilibrium point in the center. The magnitude of d2ψ/dx2— which governs how fast ψ will oscillate—is proportional to the magnitude of (E − PE). Far from the equilibrium point, the total energy is almost entirely potential energy. This makes (E − PE) very small, or zero, which means that ψ must be large in order for the product of the two, (E − PE)ψ, to be large enough for d2ψ/dx2 to make the curve “bend over.” That is why the peaks in the wave are separated more at the edges, and higher as well.
The higher up we go in discrete energy states, the more the probability density resembles the classical curve. Figure 5.16 shows |ψ12|2, corresponding to n = 12. If we averaged this function over x, it would have approximately the general shape of the classical probability,
p(x)
FIGURE 5.16 The probability distribution from quantum mechanics, |ψ|2, compared to that of classical mechanics, p(x). At the zero-point energy (n = 0), quantum mechanics predicts that the oscillator is most likely to be near the equilibrium spacing, xe. Classical mechanics predicts just the opposite: that the oscillator spends more time at the edges of its motion, near A and −A. But as n increases, both the quantum and classical probabilities begin to look similar. We can see this progression toward correspondence here, from n = 0, to n = 2, and then n = 12.
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p(x). As we reach higher and higher values of n, the curves of |ψn|2 and p(x) look more and more similar. In most macroscale systems, energy states are extremely high. (For example, a 1-milligram object traveling a few centimeters per second has an energy state, n, on the order of 1023.) Actually, in many nanoscale systems, the energy states are also very high, typically above 105 (see Back-of-the-Envelope 5.6).
Thus, it is crucial that classical and quantum mechanics do not predict completely dif-ferent behavior. The two theories need to jibe. Quantum mechanics is better suited to the nanoscale, where energy comes and goes in packets. However, these packets become more indistinguishable the more atoms there are, until eventually energy is considered continu-ous. What is important is that there be a regime in which classical and quantum mechanics see eye to eye—where their predictions overlap. This is known as the Correspondence Principle.
There is one final discrepancy to clear up. You have probably noticed that the tail ends of the probability curves actually extend beyond the rigid limits of motion predicted by classical physics. When we introduced the concept of the wave function in Chapter 3, we assigned infinite height to the boundaries of the “potential well” in which the particle moved. This meant that the wave function would have to be zero at the edges of the well.
The difference here, then, is that the boundaries are not infinitely high. Because they are finite, there is a slim chance that the particle will actually be found outside the confines of the well—or in this case, beyond the classically allowed region of oscillation. We will discuss this phenomenon in greater detail when we cover “tunneling” in Chapter 6,
“Nanoelectronics.” Here, we will simply note that these tails actually shrink in toward the limits as we go to higher values of n, until the classical and quantum pictures look practically identical.