 So far, we found a wave function such that the destruction operator applied to it produces the wave function times an arbitrary complex constant c. And, we've seen that magnitude c squared is the expected number of photons in the mode. Now, let's calculate the expectation value of the electromagnetic field operator for our wave function. Per reference, here's the recursion formula for the bn coefficients. The wave function is the sum over n of bn times quantum state n. And here we explicitly show the time dependence, e to the minus i n plus one half omega t. For the bra version, we'll use m for the summation variable. We have the bra version of quantum state m times the complex conjugates of bm and the exponential time factor. Then, the expectation value of the destruction operator is the sum over m, the sum over n of e to the minus i n minus m omega t times bm conjugate bn times bra m a hat minus ket n. One thing to notice is that the one half terms in the complex exponential factors have canceled out. In video two, we saw that these terms correspond to the zero point energies, that when summed over all modes, give us a very inconvenient infinity. This calculation shows why we can often get away with ignoring this infinity. The zero point terms cancel and so have no effect on our final result. Back to our expectation value, the destruction operator destroys the n equals zero ket and converts the general n ket into square root n times the n minus one ket. Replacing n with n plus one, we get an expression that contains as a factor the projection of the n photon state onto the m photon state. This is zero, unless m equals n, in which case it's one. Replacing m by n and dropping the m summation, we get the sum over n e to the minus i omega t bn conjugate bn plus one square root n plus one. Substituting the formula at top left for bn plus one, we end up with sum over n e to the minus i omega t c magnitude squared bn. Since some magnitude squared bn equals one, this gives us, for the expectation value of the destruction operator, c times e to the minus i omega t. A very similar analysis shows that the expectation value of the creation operator is the conjugate of this, c conjugate times e to the i omega t. Therefore, our electromagnetic field operator has an expectation value polarized in the x direction times square root one over two omega times the quantity c e to the i kz minus omega t plus the complex conjugate. Writing the complex number c as magnitude c times e to the i theta, and using the fact that a complex expression plus its conjugate equals twice the real part of the expression, the quantity in parentheses becomes two magnitude c cosine kz minus omega t plus theta. This is identical to the classical wave we started with, provided we identify two magnitude c as our original amplitude b. Now, this does not mean that the quantum field is identical to the classical field, only that, on average, a measurement of the quantum field will correspond to the classical field. In fact, we've already seen that the amplitude of the quantum field must be fuzzy in some sense, because it does not contain a precise number of photons. Instead, the number of photons is described by a probability distribution with mean value magnitude c squared. This is analogous to the quantum harmonic oscillator for which the particle position is given by a fuzzy probability distribution centered on the classical particle position. So the quantum field must be represented by a fuzzy distribution centered on the classical field value. The size of these quantum fluctuations is quantified by the field's root mean square deviation, sigma a. This is the square root of the expectation value of a squared minus the square of the expectation value of a. Following through the somewhat lengthy calculations, we find the value is square root 1 over 2 omega. This is independent of the constant c, hence of the classical field amplitude, and of the expected number of photons present in the mode. So, as field amplitude increases, quantum noise becomes an ever smaller fraction of the field. And we approach the classical case where the field has a precisely defined value throughout space and time. But as field amplitude decreases, quantum noise becomes an ever larger fraction of the field. In the limit c equals 0 for which the classical field vanishes, quantum fluctuations remain. This is the vacuum state. We see that in quantum field theory, a vacuum is not quote nothing or quote empty space. Quantum fluctuations of the field are always present. If we take this seriously, then we might expect there to be observable consequences. We will return to this question in future videos. Finally, we want to introduce an alternate representation of a quantum system that will simplify our future calculations. The representation we have used so far is called the Schrödinger picture. Here the time dependence of the system is contained in the wave function. Operators are typically time independent. This is the standard representation we use throughout the quantum mechanics series, and it's the way quantum mechanics is typically taught. In this video so far, we've been working in this representation. Our field operator has spatial dependence but no time dependence. The time dependence appears in the wave function. A different representation is the Heisenberg picture. Here time dependence is contained in operators, or the wave function is time independent. In the Heisenberg picture, our wave function has no time dependence. It's identical to the Schrödinger wave function evaluated at time t equals 0. The time dependence now appears in the field operator, through the factor e to the i kz minus omega t and its complex conjugate. These different representations lead to the same physical predictions. For quantum field theory, the Heisenberg picture is more convenient. So going forward, we will generally use this representation. As an example of the convenience of the Heisenberg picture, consider the classical field description of a single electromagnetic mode. ak alpha equals unit vector ek alpha over square root 2 omega times 2 amplitude ak alpha times cosine k dot x minus omega t plus theta k alpha. Writing this in terms of complex exponentials, we get an expression with two terms. One with amplitude ak alpha and the other with amplitude ak alpha conjugate. Comparing this to the Heisenberg field operator, they appear almost identical. The only change is that the amplitudes are replaced by destruction and creation operators. Apparently, if we have a classical field expression, we can immediately transition to a quantum field representation by this simple replacement. In the classical field expression, which amplitude factor we denote as the conjugate is arbitrary? What determines the type of operator we replace them by is the time dependence. The amplitude associated with the e to the minus i omega t factor becomes the destruction operator. While the amplitude associated with the e to the plus i omega t factor becomes the creation operator.