Introduction to Methods of Quantum mechanics

So far, we have discussed some of the experimental evidence that led to the breakdown of classical physics and to the beginning of quantum mechanics. We have seen how the introduction of the quantization postulates explained the experimental facts concerning blackbody radiation, the photoelectric effect, and the hydrogen spectrum. These theories constitute what we call today the old quantum theory (OQT).

Despite its successes, the OQT has some serious deficiencies:

1. The theory can be applied only to periodic systems (harmonic oscillators, circular motion, and such), although there are many important physical systems that are not periodic.
2. Although the Bohr theory predicts the observed wavelengths of the spectrum of hydrogen, it does not explain why certain wavelengths are more intense than others; that is, it does not account for the rate of transition between different energy levels.
3. The Bohr theory explains well the spectrum of monatomic hydrogen (H), singly ionized helium (He +), and reasonably well those of the alkali elements (lithium Li, sodium Na, potassium K, . . .), but only because these are H-like atoms. It fails to explain the spectrum of even the simplest of the multi-electron atoms, He.
4. But perhaps the most serious criticism of the OQT is that it is intellectually unsatisfying; it is not a unified or general theory. It assigns to microscopic particles a well-defined path that, by the uncertainty principle, is not possible.

Interpretation of Wave Function

The guiding wave is represented by a mathematical function, Ψ (r, t), called a wave function. The physical significance of the wave function is the following: At some instant t, a measurement is made to locate the particle associated with the wave function Ψ.

The probability P(r, t) dV that the particle will be found within a small volume dV centered around a point with position vector r (with respect to a prechosen set of coordinates) is equal to |Ψ|2 dV, that is,

P(r, t) dV = |Ψ|2 dV

We should note that the probability of finding the particle somewhere in space must be unity. Therefore, the wave function Ψ must be normalized; that is, it must satisfy the condition

Equation

where the integration is carried over all space.The methods of quantum mechanics consist in first finding the wave function associated with a particle or a system of particles

Characteristics of the wave function

1. The wave function should have a finite value at every point. it should not has infinite value as shown in fig a
2. Wave function should have only one single value at the point and cannot have many values as shown in fig b
3. the function should be continuous. it should not be discontinuous as shown in fig c