PHYSICAL INTERPRETATION AND
CONDITIONS ON ψ
Suppose we have set up the Schrödinger equation for a particle. Can we say that each
and every solution of the equation describes a possible matter wave associated with the
particle?
The answer is ‘no’. Only those solutions which satisfy certain general conditions are
admissible for the description of any physical system. The conditions arise partly from
the physical interpretation of ψ and partly from the nature of the wave equation itself.
We now turn to consider these.
Properties of wavefunction and constraints on Wavefunction
1. Must be a solution of the Schrodinger equation.
2. Must be normalizable. This implies that the wavefunction approaches zero as x
approaches infinity.
3. Wave Function is Single Valued. Since it is linear and we know by mathematically
that the linear function has one value, which means that it is single valued. In
mathematics, any function which is not linear, that contains more than two values.
The quadratic or cubic functions are not single value functions.
4. Wave Function is Finite. Since it is linear and therefore it has single value. If it
is single value then every single value is finite. This means that the wave function
is finite. If ψ is infinite for a particular point, it would mean an infinitely large
probability of finding the particle at that point. This would violet the uncertainty
principle.
5. Must be a continuous function of x. The slope of the function in x must be contin-
uous. Specifically ∂ψ/∂x must be continuous. This is one of the main properties
of wave function. Wave function must have linear mathematical representations.
The powers or the exponent of wave function is one. So, therefore it is continu-
ous. It can’t be quadratic or contains the higher exponents. Otherwise, the wave
function will lose its characteristics and it will be no more as a wave function.
We will discuss the Schrodinger wave equation in the upcoming articles, where we
will show that the wave function have linear mathematical representations. The
physical significance of Schrodinger wave equation is continuous.
1
, 2
CONDITIONS ON ψ
Suppose we have set up the Schrödinger equation for a particle. Can we say that each
and every solution of the equation describes a possible matter wave associated with the
particle?
The answer is ‘no’. Only those solutions which satisfy certain general conditions are
admissible for the description of any physical system. The conditions arise partly from
the physical interpretation of ψ and partly from the nature of the wave equation itself.
We now turn to consider these.
Properties of wavefunction and constraints on Wavefunction
1. Must be a solution of the Schrodinger equation.
2. Must be normalizable. This implies that the wavefunction approaches zero as x
approaches infinity.
3. Wave Function is Single Valued. Since it is linear and we know by mathematically
that the linear function has one value, which means that it is single valued. In
mathematics, any function which is not linear, that contains more than two values.
The quadratic or cubic functions are not single value functions.
4. Wave Function is Finite. Since it is linear and therefore it has single value. If it
is single value then every single value is finite. This means that the wave function
is finite. If ψ is infinite for a particular point, it would mean an infinitely large
probability of finding the particle at that point. This would violet the uncertainty
principle.
5. Must be a continuous function of x. The slope of the function in x must be contin-
uous. Specifically ∂ψ/∂x must be continuous. This is one of the main properties
of wave function. Wave function must have linear mathematical representations.
The powers or the exponent of wave function is one. So, therefore it is continu-
ous. It can’t be quadratic or contains the higher exponents. Otherwise, the wave
function will lose its characteristics and it will be no more as a wave function.
We will discuss the Schrodinger wave equation in the upcoming articles, where we
will show that the wave function have linear mathematical representations. The
physical significance of Schrodinger wave equation is continuous.
1
, 2
