The waves associated with a material particle are called matter waves. Wavelength of matter wave associated with a particle of momentum p is given by,
l=h/p --------- (1)
l is called de-Broglie wavelength.
Newtonian mechanics and quantum mechanics
In Newtonian mechanics or classical mechanics, the future state of a particle is completely determined by its initial position and momentum together with the forces that act upon it. In contrast to the above, the structure of quantum mechanics is built upon the foundation of particles which are purely probabilistic in nature. Thus instead of asserting, for example, the radius of electron orbit in a ground state of hydrogen atom is always exactly 0.53Å, quantum mechanics states that this is the most probable radius.
Quantum mechanics is related to microscopic universe while Newtonian mechanics deals with macroscopic universe.
Newton’s second law of motion is the basis of Newtonian mechanics while Schrödinger equation is the basis of quantum mechanics.
Heisenberg’s uncertainty principle
According to classical mechanics, it is possible for a particle to occupy a fixed position and have a definite momentum and we can predict exactly its position and momentum at any time later. But according to uncertainty principle, it is not possible to determine accurately the simultaneous values of position and momentum of a particle at any time. Heisenberg’s principle implies that in physical measurements, probability takes the place of exactness and as such phenomena which are impossible according to classical ideas may find a small but finite probability of occurrence.
Statement
It is impossible to determine precisely and simultaneously the values of both the members of a pair of physical variables which describe the motion of an atomic system. Such pairs of variables are called canonically conjugate variables.
Examples:1) Position and momentum
2) Energy and time
Wavefunction
The quantity with which quantum mechanics is concerned is the wavefunction y of a particle. We can obtain all the physical properties of a system if we know the wavefunction. The quantity y2 or y*ydxdydz is proportional to the probability of finding the particle in the volume element dxdydz about the point (x,y,z).
If òy*ydxdydz=0, the particle does not exist.
If òy*ydxdydz=a, the particle is everywhere simultaneously
Since the particle exist somewhere at all times,
òy*ydxdydz=1 --------- (2)
The wavefunction y satisfying the above condition is called normalized wavefunction.
Requirements of wavefunction
The wavefunction must be continuous and single valued everywhere.
∂y/∂x, ∂y/∂y and ∂y/∂z must also be continuous and single valued everywhere.
y can be normalized.
Wave equation
A wave whose variable quantity is y that propagates in the x-direction with speed v is expressed by an equation,
∂2y/∂x2=1/v2 ∂2y/∂t2 --------- (3)
This is called wave equation.
Schrödinger’s equation for a free particle-time dependent equation
Schrödinger’s equation is a wave equation with the variabley. It is the fundamental equation of quantum mechanics in the same sense that the second law of motion is the fundamental equation of Newtonian mechanics.
The quantity with which quantum mechanics is concerned is the wavefunction y of a particle. We can obtain all the physical properties of a system if we know the wavefunction. The quantity y2 or y*ydxdydz is proportional to the probability of finding the particle in the volume element dxdydz about the point (x,y,z).
If òy*ydxdydz=0, the particle does not exist.
If òy*ydxdydz=a, the particle is everywhere simultaneously
Since the particle exist somewhere at all times,
òy*ydxdydz=1 --------- (2)
The wavefunction y satisfying the above condition is called normalized wavefunction.
Requirements of wavefunction
The wavefunction must be continuous and single valued everywhere.
∂y/∂x, ∂y/∂y and ∂y/∂z must also be continuous and single valued everywhere.
y can be normalized.
Wave equation
A wave whose variable quantity is y that propagates in the x-direction with speed v is expressed by an equation,
∂2y/∂x2=1/v2 ∂2y/∂t2 --------- (3)
This is called wave equation.
Schrödinger’s equation for a free particle-time dependent equation
Schrödinger’s equation is a wave equation with the variabley. It is the fundamental equation of quantum mechanics in the same sense that the second law of motion is the fundamental equation of Newtonian mechanics.
Eigen values and Eigen functions
By solving the Schrödinger equation, we obtain the possible set of y functions. In case of bound particles, the acceptable solutions for the differential equations are possible only for certain specified values of energy. These descrete values of energy E1,E2,........,En are called energy eigen values of the particle. The solutions y1,y2,...........,y3 corresponding to the eigen values are called eigenfunctions.
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