Wavefunction Normalisation

According to the Born interpretation, the probability of locating a particle within a volume interval is,

`"probability"=|Psi(x,y,z,t)|^2deltaV`

By definition, the probability of locating the particle somewhere in all space is definite - ie., the particle is definitely somewhere. This leads us to the normalisation condition,

`int_(-oo)^oo |Psi(x,y,z,t)|^2dV=1`

A wavefunction that satisfies this condition is said to be normalised. Whilst a wavefunction may satisfy the wave-equation, it is not necessarily normalised. To normalise a wavefunction, we introduce a coefficient, namely, the normalisation coefficient, which scales the wavefunction in such a way that the normalisation criteria is satisfied. Since the Hamiltonian is a linear operator, then if, `Psi(x,y,z,t)` satisfies the wave-equation, then so does `APsi(x,y,z,t)`.

For any wavefunction, where the integral,

`I=int_(-oo)^oo |Psi(x,y,z,t)|^2dV!=1`

we can introduce a normalisation coefficient to scale I to equal one. Hence,

`1=int_(-oo)^oo |APsi(x,y,z,t)|^2dV=A^2int_(-oo)^oo |Psi(x,y,z,t)|^2dV`

Thus we can express the normalisation coefficient as,

`A=1/sqrt(int_(-oo)^oo |Psi(x,y,z,t)|^2dV)=(int_(-oo)^oo |Psi(x,y,z,t)|^2dV)^(-1/2)`

Example I: Normalisation of a Linear Combination of Atomic Orbitals:

Consider a homonuclear diatomic bonding molecular orbital, `Psi(x,y,z,t)_+` composed of two atomic orbitals, `phi_1(x,y,z,t)` and `phi_2(x,y,z,t)`. For the sake of clarity we omit the wavefunction arguments, but it is to be understood that `Psi_+=Psi(x,y,z,t)_+` and `phi_j=phi(x,y,z,t)_j`. Note, `d tau` represents a volume. From LCAO theory we have,

`Psi_+=c_1phi_1+c_2phi_2`

Since the MO is homonuclear, each atom contributes an equal weighting to the resultant combination, hence `c_1=c_2=c_+`.

`Psi_+=c_+(phi_1+phi_2)`

For the LCAO combination, the normalisation criteria requires,

`1=int_(-oo)^oo |Psi_+|^2d tau`

Using the LCAO composition of `Psi_+`, we have,

`1=c_+^2int_(-oo)^oo |phi_1+phi_2|^2d tau`

Where the atomic orbitals are complex, the modulus expands to give,

`1=c_+^2int_(-oo)^oo (phi_1^**+phi_2^**)(phi_1+phi_2)d tau`

We expand the brackets and distribute the integration,

`1=c_+^2(int_(-oo)^oo phi_1^**phi_1d tau+int_(-oo)^oophi_2^**phi_1d tau+int_(-oo)^oophi_1^**phi_2d tau+int_(-oo)^oophi_2^**phi_2d tau)`

We understand that each of the atomic orbital wavefunctions are already normalised, such that,

`1=int_(-oo)^oo |phi_j|^2d tau=int_(-oo)^oo phi_j^**phi_jd tau`

Also that the overlap integrals are close to zero,

`0~~int_(-oo)^oo phi_1^**phi_2d tau=int_(-oo)^oo phi_2^**phi_1d tau`

Hence

`1=c_+^2(1+0+0+1)=2c_+^2`

...and we arrive at the normalisation coefficients,

`c_+=1/sqrt(2)`

Thus the normalised expression for the homonuclear diatomic bonding orbital is,

`Psi_+=phi_1/sqrt(2)+phi_2/sqrt(2)`

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