# Wave function and its physical significance

**WAVE FUNCTION**

If there is a wave associated with a particle, then there must be a function to represent it. This function is called wave function.

Wave function is defined as that quantity whose variations make up matter waves. It is represented by Greek symbol ψ(psi), ψ consists of real and imaginary parts.

Ψ=A+iB

**PHYSICAL SIGNIFICANCE OF WAVE FUNCTIONS (BORN’S INTERPRETATION):**

** Born’s interpretation**

** **The wave function ψ itself has no physical significance but the square of its absolute magnitude |ψ^{2}| has significance when evaluated at a particular point and at a particular time |ψ^{2}| gives the probability of finding the particle there at that time.

The wave function ψ(x,t) is a quantity such that the product

P(x,t)=ψ^{*}(x,t)ψ(x,t)

Is the probability per unit length of finding the particle at the position x at time t.

P(x,t) is the probability density and ψ^{*}(x,t) is complex conjugate of ψ(x,t)

Hence the probability of finding the particle is large wherever ψ is large and vice-versa.

**NORMALIZATION CONDITION**

The probability per unit length of finding the particle at position x at time t is

P=ψ^{*}(x,t)ψ(x,t)

So, probability of finding the particle in the length dx is

Pdx=ψ^{*}(x,t)ψ(x,t)dx

Total probability of finding the particle somewhere along x-axis is

∫pdx =∫^{ }ψ^{*}(x,t)ψ(x,t)dx

If the particle exists , it must be somewhere on the x-axis . so the total probability of finding the particle must be unity i.e.

∫ψ^{*}(x,t)ψ(x,t)dx=1 (1)

This is called the normalization condition . So a wave function ψ(x,t) is said to be normalized if it satisfies the condition(1)

**ORTHOGONAL WAVE FUNCTIONS**

Consider two different wave functions ψ_{m} and ψ_{n }such that both satisfies Schrodinger equation.These two wave functions are said to be orthogonal if they satisfy the conditions.

Or ∫ ψ_{n}^{* }(x,t) ψ_{m}(x,t) dV=0 for n≠m] ( 1)

∫ ψ_{n}^{* }(x,t) ψ_{m}(x,t) dV=0 for m≠n ]

If both the wave functions are simultaneously normal then

∫ ψ_{m} ψ_{m}^{*} d V=1=∫ψ_{n}ψ_{n}^{*} dV (2)

**Orthonormal wave functions:**

The sets of wave functions, which are both normalized as well as orthogonal are called orthonormal wave functions.

Equations (16) and (17) are collectively written as

∫ψ^{*}_{m}ψ_{n}dV={ o if m≠n

=[1 if m=n

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we want more information

like??

like considerin a two particle like electrons or some others and assosciate the wave function and put them in to debate of normailizatn

is normalizion of wave function possible to explain physically