Wave function is a term that is
used in the quantum mechanics to describe the wave in a mathematical form. It
explains the quantum state in each quantum isolated system. Wave function is
denoted by a Greek letter psi Ψ. The value of a particle's wave function at a
particular place in space and time is proportional to the probability of the
particle being there at that moment. Ψ2 the square of wave function
is physically significant it gives us the probability density of a particle at
a given space and time.
The wave function is a function
of degrees of freedom that correspond to a maximum set of commuting
measurements. The wave function may be deduced from the quantum state once such
a representation is chosen.
As we know that Ψ is a wave
function and according to Bohr Ψ2 does not measure the particles
density at any point and gives the probability of finding that particle at that
point in each moment.
Ψ2 = Ψ Ψ*
Here, Ψ is the real wave function
and Ψ* is the imaginary wave function. The more exactly we can say that the
probability of the particles being present in the volume dxdydx is Ψ2
dxdydz, we can write it as,
Ψ2 dxdydz =1
Here, the wavefunction Ψ
satisfying this requirement is said to be normalized.
Ψx2 dx = 1 is called
as one-dimensional normalization the wave equation.
Ψx2 dxdy = 1 is called
as two-dimensional normalization the wave equation.
Ψx2 dxdydz= 1 is
called as three-dimensional normalization the wave equation.
The wavefunction Ψ has some
limitations to it, which are as follow
1-
Ψ must be finite to all the values of x, y and z
2-
Ψ must be single valued such as for each set of
values of x, y and z, y must have only one value.
3-
Ψ must be in all regions except where potential
energy is infinite
4-
Ψ is analytical it possesses continuous first
order derivative
5-
Ψ vanishes at the boundaries
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