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,  x,t   We wish to answer the following questions:  Where is exactly the particle located within  x? 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   is proportional to the intensity of the wave). Now, what does |   physically mean?

3 Probabilistic interpretation of (the square of) matter wave  As seen in the case of radiation field, |electric field’s amplitude   is proportional to the probability of finding a photon  In exact analogy to the statistical interpretation of the radiation field,  P(x) = |   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

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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 Bothequantum mechanicsabsolute squareSchrödinger equationBothe

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 = = constant) L = 1 /2 (n = 1) L =  (n = 2) L = 3 3 /2 (n = 3) L