As we mentioned previously, the motion of a string is, in general, very complicated.
However, there are so-called characteristic modes of the string that are relatively simple
to describe. These have the form of a single function of time, q-k of t , times a spatial
sine wave, sine k pi over L x, where k is a positive integer.
Plugging this expression into the wave equation, we get q-k double-dot times the sign, equals
minus c-squared times q-k times the sine. Canceling the common sine factor we are left
with, q-k double-dot equals minus quantity c k pi over L squared times q-k.
Defining the quantity in parenthesis as omega-k we arrive at the equation of motion for this
k-th mode. q-k double-dot equals minus omega-k squared times q-k.
We recognize this as identical to the equation of motion for the harmonic oscillator, that
we described in detail in the previous video. These modes form the basis of the Fourier
series method for describing an arbitrary oscillation of the string. The k-th mode has
sinusoidal spatial dependence with k bumps. The first mode, with a single bump, oscillates
at a low frequency. The second mode, with two bumps, oscillates
at twice that frequency. And the k-th mode, with k bumps, oscillates
at k times the first mode frequency. If you sum all the modes you get a complicated,
non-sinusoidal oscillation of the string. In fact any physically possible string oscillation
can be represented by a superposition of these simple modes with suitably chosen amplitudes
q-k . Each of which satisfies the harmonic-oscillator
equation of motion at frequency omega-k.
Therefore, an arbitrary oscillation of the string is mathematically equivalent to the
oscillations of an infinite number of independent harmonic oscillators, each with its own oscillation
frequency. Since we know how to treat a single harmonic
oscillator quantum mechanically, this should allow us to treat any field quantum mechanically
by simply combining the independent contributions of each member of this infinite set of harmonic
oscillators. Based on this idea we can immediately write
down the wavefunction of a quantum string. The classical system represents the string
oscillation as the superposition of sinusoidal modes. The amplitudes of which satisfy the
harmonic oscillator equation of motion for their respective frequency.
The wavefunction for a single mode has the characteristics we established in the previous
video. psi-k of q-k is proportional to a polynomial H-n-k of xi-k times e-to-the minus one half
xi-k squared. Where xi-k is proportional to the amplitude q-k.
The mode energy is E-k equals quantity n-k plus one half times h-bar omega-k, where n-k
is the number of energy quanta contained in the mode.
The wave function of the entire string system is, psi of q-1, q-2, q-3, etc. equals psi-1
of q-1 times psi-2 of q-2 times psi-3 of q-3 etc.
Or using the Dirac ket notation, the quantum state of the system is described by the occupation
numbers n-1, n-2, n-3, etc. where n-k is the number of energy quanta in the mode with frequency
omega-k. The modes oscillate independently, so the quantum state of the entire system
is the product of the quantum states of the individual modes. In applications this turns
out to be the more useful representation. We can form creation
And destruction operators for each mode. Combine those into a mode number operator.
And use that to extract the number of energy quanta in that mode.
When applied to the complete wavefunction, these operators act only on the part corresponding
to their respective mode. This allows us, for example, to create or destroy energy quanta
in specific modes while not affecting other modes.
Or, to operate on the complete wavefunction and extract the energy of a single mode.
If we want to we can make use of the probability interpretation of the wavefunction. The magnitude
squared of the string’s wavefunction equals the magnitude squared of the first-mode wave
function times the magnitude squared of the second-mode wavefunction and so on.
This gives us the probability density that the amplitude of mode 1 is q-1, the amplitude
of mode is q-2, etc. And, the energy of the entire system is the
sum of all mode energies. n-k plus one-half times h-bar omega-k.
Where n-k is the number of energy quanta in mode k . When we apply these ideas to the
electromagnetic field, n-k will be the number of photons with frequency omega-k.
So there we have it. A quantum field theory of a string. Now, it doesn’t do much, since
it’s not interacting with other systems. We have mechanisms to create or destroy energy
quanta, but nothing to set those in motion. The system just sort of sits there and does
nothing. Still, it’s our first step. All we need to do now is to extend this idea to
various types of fields and come up with a way to describe their interactions.
But, we already have a problem. Note that even if none of the modes contain energy quanta
they each still have non-zero energy, one-half h-bar omega-k. And, there are an infinite
number of modes. So the lowest energy of the system, the zero-point energy, is infinite.
How could this possible be? The non-zero, lowest-energy of a harmonic
oscillator is required by the uncertainty principle. A mass cannot be at a known position
with a known momentum. So a quantum mass on a spring cannot be perfectly at rest, hence
it cannot have zero energy. Now, a real string, or any physical material,
is actually not a continuous system. Instead, it’s made up of a finite number of discrete
masses – its individual atoms. A finite number of masses can only support a finite
number of modes, so the zero-point energy is not actually infinite.
But, when we apply these ideas to non-mechanical, continuous systems, such as the electromagnetic
field, this argument is not available. We might argue that zero-point energy has no
observable effects, since we can only measure energy differences, and then just ignore it.
One text sums this up as, “We declare by fiat that the ground state has zero energy.”
Regardless of the argument, we can get rid of the infinite zero-point energy by only
taking into account energy contributions from the quanta of the various modes.
We’ll see that this works, but has a distinct hand-waving quality to it. In fact, this won’t
be the last infinity we have to deal with in quantum field theory.