Recap: statistical interpretation of radiation

1. Recap: statistical interpretation of radiation • The probability of observing a photon is proportional to N (the number of photon crossing a unit cross section in a unit time) • Also, correspondence principle says • Hence the probability of observing a photon is Prob (x) Square of the mean of the square of the wave field amplitude

2. What is the physical interpretation of matter wave? • we will call the mathematical representation of the de Broglie’s wave / matter wave associated with a given particle (or an physical entity) as The wave function, Y(x,t) • We wish to answer the following questions: • Where is exactly the particle located within Dx? the locality of a particle becomes fuzzy when it’s represented by its matter wave. We can no more tell for sure where it is exactly located. • Recall that in the case of conventional wave physics, |field amplitude|2 is proportional to the intensity of the wave). Now, what does |Y |2physically mean?

3. Probabilistic interpretation of (the square of) matter wave • As seen in the case of radiation field, |electric field’s amplitude|2 is proportional to the probability of finding a photon • In exact analogy to the statistical interpretation of the radiation field, • P(x) = |Y |2 is interpreted as the probability density of observing a material particle • More quantitatively, • Probability for a particle to be found between point a and b is

5. Hence, a particle’s wave function gives rise to a probabilistic interpretation of the position of a particle • Max Born in 1926 • German-British physicist who worked on the mathematical basis for quantum mechanics. Born's most important contribution was his suggestion that the absolute square of the wavefunction in the Schrödinger equation was a measure of the probability of finding the particle at a given location. Born shared the 1954 Nobel Prize in physics with Bothe

6. Some weird philosophical inferences of the probabilistic interpretation • Due to the probabilistic interpretation of the matter wave, the notion of “existence” of a physical entity, at its most fundamental level, begins to deviate from our conventional wisdom • The existence of an entity is now no more be deterministic notion (e.g. it either exist or not at all) but only a “probability” • If interested, please read the philosophical interpretation of quantum mechanics yourself • Its one of the most intriguing argument of the last century and is still continue to be so

7. Quantum description of a particle in an infinite well • Imagine that we put particle (e.g. an electron) into an “infinite well” with width L (e.g. a potential trap with sufficiently high barrier) • In other words, the particle is confined within 0 < x < L

8. Another experimental scenario to trap a particle: using electric potential trap: As V  infinity, the potential trap approaches that of an idealised infinite quantum well The charged particle moves freely inside the region of zero potential, 0 < x < L. But it would be bounced back when it bangs on the infinitely “hard” wall. Mathematically this is described by saying that the potential takes the form

9. Particle forms standing wave within the infinite well • How would the wave function of the particle behave inside the well? • They form standing waves which are confined within 0 ≤x≤ L

10. Standing wave in general • Description of standing waves which ends are fixed at x = 0 and x = L. (for standing wave, the speed is constant), v = ln= constant) L = l1/2 (n = 1) L = l2 (n = 2) L = 3l3/2 (n = 3) L