Nano Materials by A.K. Bandyopadhyay

By A.K. Bandyopadhyay

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Let us start with a vibrating object → For an object attached to a spring, Hook’s law describes the system in terms of the force (F) on the object and the displacement (x) from the equilibrium position as: F = – kx where k = Spring Constant or Force Constant (Newton. mt or Dynes/cm) The negative sign means that the resting force or spring tension is in the direction opposite to the displacement. e.  N @N = N ∫ ⎛1⎞ – kx dx = ⎜ ⎟ kx2 ⎝ 2⎠ If the mass (m) is displaced by a distance of x and released, the object vibrates in simple harmonic motion.

The coordinate x is simply used in operator form as x. Some operators can be combined, formed. , since the kinetic energy is p2 mV 2 , it can be written in terms of the momentum p, as . 2m 2 The operators that are important in quantum mechanics have two important characteristics : 1. 19) where α is the operator and φ1 and φ2 are the ‘functions’ being operated on. 20) The linear character of the operator is related to the superposition of 'states' and waves reinforcing each other in the process.

Finite. A probability of unity denotes a ‘sure thing’. A probability of zero means that a particular event can not happen. Hence, the probability varies from zero to unity. If ψ were infinite, the probability could be greater than unity. 2. Single valued. In a given area of space, there is only one probability of finding a particle. For example, there is a single probability of finding an electron at some specified distance from the nucleus in a hydrogen atom. There can not be two different probabilities of finding the electron at some given distance.

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