 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, qk 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 qk double dot times the sine equals minus c squared times qk times the sine. Canceling the common sine factor, we are left with qk double dot equals minus quantity ck pi over l squared times qk. Defining the quantity and parentheses as omega k, we arrive at the equation of motion for this kth mode. qk double dot equals minus omega k squared times qk. 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 kth 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 kth mode, with k bumps, oscillates at k times the first mode frequency. If you sum all the modes, you get a complicated, non-s 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 qk. 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 wave function 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 the respective frequency. The wave function for a single mode has the characteristics we established in the previous video. Psi k of qk is proportional to a polynomial hnk of psi k times e to the minus one half psi k squared, where psi k is proportional to the amplitude qk. The mode energy is ek equals quantity nk plus one half times h bar omega k, where nk is the number of energy quanta contained in the mode. The wave function of the entire string system is psi of q1, q2, q3, etc. Equals psi1 of q1 times psi2 of q2 times psi3 of q3, etc. Or, using the Dirac ket notation, the quantum state of the system is described by the occupation numbers n1, n2, n3, etc., where nk 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 wave function, these operators act on only the part corresponding to their respective mode. This allows us, for example, to create or destroy energy quanta in specific modes while not affecting the other modes. Or, to operate on the complete wave function and extract the energy of a single mode. If we want to, we can make use of the probability interpretation of the wave function. The magnitude squared of the string's wave function equals the magnitude squared of the first mode wave function times the magnitude squared of the second mode wave function, and so on. This gives us the probability density that the amplitude of mode 1 is q1, the amplitude of mode 2 is q2, etc. And, the energy of the entire system is the sum of all mode energies, nk plus 1 half times h bar omega k. nk is the number of energy quanta in mode k. When we apply these ideas to the electromagnetic field, nk 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, 1 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 possibly 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 strength, 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 up this approach 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.