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Wave Function and its Physical Interpretation

The wave function ψ (r, t) explains the behavior of the particle at a given position r at a given  time. The magnitude of wave function is large in regions where probability of finding the particle is high and its magnitude is small in regions where the probability of finding the . particle is low. Therefore, ‘If may be taken as a measure of the probability of the particle around a particular position. It is already assumed that the wave associated with a particle in motion may be

represented by a complex variable quantity ‘lf(x, y, z, t) is known as the wave function. Since it is complex quantity, it may be expressed as

 

ψ (x ,y ,z ,t) =a +ib

where a arid b are real functions of variables (x, y, z, t) and i = √(-1) .

 

The complex conjugate of ‘If is written as ψ* can be obtained by changing· i to – i. ·

Therefore

Ψ* (x , y, z, t) =a =ib

 

Multiplying equation (1) by equation (2), we get

 

Ψ* (x , y, z, t) Ψ* (x , y, z, t) = a2 – i2 b2

Ψ* (x , y, z, t) Ψ* (x , y, z, t) = a2 + b2

 

We may write the product on left hand side of equation (3)

P =  |Ψ* (x , y, z, t)|2

|ψ* (x , y, z, t)|2 = a2 +b2

 

Therefore, the product of  ψ and .ψ* is real and positive if ‘ψ = 0. Its positive square root is denoted by ψ ( X, y, z, t) I· It represents the modulus of ‘If·

 

The quantity  ψ(x, y, z, t) 12 is called the probability density i.e., probability per unit

volume.

 

9.6.1 Max Born  Interpretation of V

 

Max Born used equation (4) to give the interpretation of ψ (x, y, z, t), which is as follows :

Consider a small element of volume dV defined by the coordinates (x, x + dx), (y, y + dy) and (z, z + dz) or (r, r + dr).

 

Then the probability of finding the particle existing within the element of volume dV

is given by

 

p d V  =Ψ* (x , y, z, t)  Ψ* (x , y, z, t) dV

|Ψ* (x , y, z, t)| 2 dV

 

For the motion of a particle in one-dimension the quantity

 

Pdx = Ψ* (x , t) Ψ* (x , t) = | Ψ* (x , t) |

 

is the probability that the particle will be found over a small distance dx at position x at

time t. | Ψ (x ,t) 12 is called the probability per unit distance.

 

Ψ* (x , y, z, t) = u Ψ* (x , y, z, t)

Ψ* (x , y, z, t) = u Ψ* (x , y, z, t)

PdV = U Ψ* (x , y, z, t) . u* (x , y, z,)dV

= | u (x, y, z)2n Dv

 

Therefore, the wave function, which satisfies the time independent Schrodinger equation, the probability is always independent of time. ·

 

9.6.2 Limitations of ψ

 

The solutions of ‘I’ obtained from the Schrodinger equation are subject to following limitations :

(i) ψ must be finite for all values x, y, z.

 

(ii) ψ must be single-valued function i.e., for each set of values of x, y, z ; ψ must have one value only.

 

(iii) ψ must be continuous in all regions except in those regions where the potential energy V(x, y, z) = ∞

 

9.6.3 Normalisation of Wave  Function

 

The solution of three dimension time independent Schrooinger wave equation for a particle gives, an expression for the-wave function of particle, which is function of (x, y, z) and contains an infinite constant. The value of the constant along with its sign may be determined as follows

 

Since ψ (x, y, z) 2 dV is the probability that the ,particle will be found in a volume

element dV surrounding the point at position (x. y, z), the total probability that the particle

will be somewhere in space must be equal to 1.

 

Thus, we have_

| Ψ (x, y, z)|2 dV = 1

where Ψ is a function of space coordinates (x, y, z)

From this normalising condition, we can find the value of the constant and its sign. A wave function which satisfies the above condition is said to be normalised (to unity).

The normalizing condition for the wave function for the motion of a particle in one dimension is |Ψ(x)|2 dx = 1