quantum mechanics Mechanics in which the dynamical variables such as momentum, energy, etc., of a particle may not form a continuous set.

The quantum mechanics has several domains of application. (i) Non relativistic quantum mechanics - when the speeds of particles are much less compared to the speed of light. (ii) Relativistic quantum mechanics - applicable to single relativistic particle. Particles may have zero rest mass and travel with speed of light. (iii) Quantum field theory - creation and destruction of particles may occur.

In classical physics the observable characterizing a given system can be simultaneously measurable in principle with arbitrary small error. The uncertainty principle excludes such possibility in quantum mechanics. Accurate measurement of an observable quantity necessarily produces uncertainties in one’s knowledge of the value of other observable. The product of two uncertainties in any two pair of variables such as position and momentum, or, time and energy is at least h/2p .

The electromagnetic radiation has wave as well as particle characteristics. The particles such as electron, neutron, proton, etc., also have wave characteristic (de Broglie hypothesis*). The dual characteristic of particle and wave is revealed exclusively of one another. This is called the Bohr’s complementary principle*.

E. Schrodinger in 1924 wrote the wave equation for the matter waves. The amplitude of matter wave is described by a complex quantity called the wave function, Y (x,y,z,t). The theory assumes the following postulates.

Postulate I Wave function Y (x,y,z,t) describes any physical system. Y (x,y,z,t) is well-behaved function, that is it is finite, single valued and continuous everywhere. This is true only if the following boundary conditions are satisfied.

(I) Y (x,y,z,t) must be continuous.

(ii) Derivatives of Y (x,y,z,t) with respect to space co-ordinates must also be continuous except where potential V(x,y,z,t) is infinite.

(iii) At infinity Y (x,y,z,t) should vanish.

(iv) in addition Y *Y dx dy dz is the probability of observing a particle in volume element dx dy dz. The total probability,

ò Y *Y dx dy dz = 1 (normalization condition) (q1)

Postulate II There is an operator to every observable quantity. The choice of the operator is arbitrary, but it must satisfy the eigenvalue equation

OY = oY (q2)

Y is the wavefunction called the eigenfuntion of the equation. O is the operator corresponding to the observable. The equation (q2) gives a set of eigenvalues, o. The result of a measurement of the observable O on the system gives a value belonging to the set.

Table QI gives a few quantum mechanical operators corresponding to different classical observable quantities.

Table QI. Quantum mechanical operators for some observable quantity

Postulate III The average value, or the expectation value, of an observable quantity o corresponding to the operator O of a physical system characterized by the wave function Y (x,y,z,t) is given by,

o = ò Y * O Y dx dy dz (q3)

where Y * is the complex conjugate of Y .

The Schrodinger wave equation: The total energy, E of a particle of mass m, and having potential energy V(x,y,z) at time t is given by,

E = p²/(2m)+ V(x,y,z)

Replacing the observable quantities p and E by their corresponding operators (table QI), and rewriting the eigenvalue equation, (q2),

- h²/(2m) Ñ ² Y + V(x,y,z,t) Y = ih[¶ /¶ t]Y (q4)

This is the time dependent Schrodinger wave equation. When the potential is time independent, the equation (q4) reduces to time independent form,

- h²/(2m) Ñ ²Y + Vy = Ey (q5)

where y is the space dependent part of the total wave function Y .

Y = y (x,y,z)T (t)

= y (x,y,z)exp{ -iEt/h}

Schrodinger applied this equation to the problem of hydrogen atom and obtained the energy levels of the electron. The Schrodinger equation has been applied to variety of problems in different branches of Physics, like spectroscopy, solid state physics, nuclear physics, etc. The theory is immensely successful in explaining various quantum phenomena.

There was a parallel theory by Heisenberg, with the operators as matrices. This theory is called matrix mechanics. This theory however has less applicability.

The wave mechanics was later extended to incorporate the relativistic effects by Dirac.