Needless to say, this is a very large error in the speed. For comparison, a free electron at room temperature would have a speed of approximately so the error imposed by a measurement of position of the given accuracy is greater than typical thermal velocities. The first parenthesis involves only terms which depend on x , the second only terms which depend on y . The next line would have an energy proportional to which is bigger than the energy of the first line, so the next line in the spectrum should be found at . The energy of a photon is related to its frequency by so in our case, The expectation value of an observable G is computed by .
All operators corresponding to observable are Hermitian.
Applying this property to the expectation value, we get The two inner products are the same, so we are saying that the expectation value must be equal to its complex conjugate. The only way that can be true is if the imaginary part is zero, so the expectation value is real.