Angular Momentum - Atomic Structure | Physical Chemistry PDF Download

ANGULAR MOMENTUM

The classical definition of angular momentum L= r X p depends on six numbers r_x, r_y, p_x, p_y and  p_z

Translating this into quantum mechanical terms, the Heisenberg uncertainty principle tells us that "it is not possible for all six of these numbers to be measured simultaneously with arbitrary precision ."

Therefore , there are limits to what can be known or measured about a particles angular momentum . It turns out that the best that one can do is to simultaneously measure both the angular momentum vectors magnitude and its component along one axis. 

Mathematically, angular momentum in quantum mechanics is defined like momentum not as a quantity but as an operator on the wave function.
 L= r X p

Where and p are position and momentum operators respectively.

Q.    Show that  

Proof: 

The II^nd term become zero because the operator is commute. A function that are dependent on three
co-ordinates x,y,z is always given the value zero because commutation  is possible in three inter dependent co-ordinate.

Ans.

VALANCE BOND THEORY

 Valance Bond Theory was the first quantum mechanical theory of chemical bond proposed by Heither and London in 1927, Slater and Pauling in 1930. It is collectively known as HLSP or HSPL. The theory propose that:

Covalent bond is formed by the shared pair of electron, i.e. Lewis concept. The theory makes the perfect pairing of approximation in which it is assumed that structures with electron paired in all possible way dominated the wave function of the molecule.

Let the 2 electron of H₂ designated by (1) and (2) and the 2 nuclei a and b. the wave function of the molecule:

ϕ are the atomic orbitals;ψ₁ is the wave function corresponding to situation, where electron (1) on the atomic orbital  centered on the nucleus (a) and electron (2) is on the atomic orbital  centered on nucleus B. Heilter and London remove C₁ and C₂  by putting a common value of N as normalization function.

Bonding M.O.=

Antibonding M.O.=

Let us normalize  we get, the value of

Similarly, we get normalization constant

Normalized wave functions:

  Q.Write down the valence bond wave function for HF molecule (assuming that it is formed from 1s orbital of H and 2p_z orbital of F) in the following three cases.

 (i)HF is purely covalent.                    (ii) HF is purely ionic.                        (iii) HF is 80% civalent and 20% ionic.

 Sol.      @

Total probability              (if inspection is 100)

 In ionic Ψ, probability=

 Operator are not fixed for an eigen value equation and a function is not always eigen function, both are independent.

Schrodinger Wave Equation

Schrodinger equation in a eigen value equation, i.e. the wave function is an eigen function and the energy is an eigen value of the hamiltanian operator.

Thus suggests a coresponce between the hamiltanian operator and the energy and this is the fundamental approach to the formation of quantum mechanism.

To prove Schrodinger equation quantum mechanically. An eigen value equation:

The eigenvalue equation:For the linear operator α , consider the equation

Where a_n is an arbitrary complex number. Equation(30) represents an eigenvalue equation;  is said to be an eigenket of the operator α, a_n being the corresponding eigenvalues. It can be easily seen that  (where c is an arbitrary complex number) is also an eigenket belonging to the same eigenvalue a_n Now, if there is more than one ket (and they are not linearly dependent on each other ) belonging to the same eigenvalue, ie. If

.............(31)
....................(32)

Then the state is said to be a degenerate state, if there are ‘g’ linearly independent kets belonging to the same eigenvalue then the state is said to be g- fold degenerate. For the sake of simplicitly, let us consider a two-fold degenerate state describe by equation (31) and (32). If we multiply equation (31) by C₁ and equation (32) by c₂ and add we would get

here  Implying that the linear combination   is also an eigenket belonging to the same eigenvalu.

Orthogonality of eigenfunctions: When α is real, it can easily be shown that all the eigenvalues are real and for two different eigenvalues  the corresponding eigenfunctions are necessarily orthogonal, i.e.

     for       ....(33)

Further, one can always normalize the kets and choose an appropriate linear combination for the kets belonging to a degenerate state such that

...........(34)   

The proof is very simple, Premultiplying equation (30) by   we get

Now,  is always real and not a null ket [otherwise equation (30) has no meaning]. Further, since α is real

 Implying that  is also real and hence a_n must be real. Father, in order to prove equation (33) we consider.

        ................(35)

Also,    ..............(36)

If we put   , then

Also,        

Because a₂ is real. Thus

 ......(37)       

Premultiplying equation (35) by  and postmultiplying equation (37) by gives

Which immediately gives the orthogonality given by equation (33)

 Since the formalism is symmetrical with respect to bras and kets we also have the iegen value equation

Where  are the eigenbras and b_n the corresponding eigenvalues. It can be easily seen that when α is a real operator and if  is an eigenket, then  is an eigenbra belonging to the same eigenvalue. Equation (37) tells us that is an eigenbra of the operator α belonging to the same eigenvalue a₂ .

The completeness condition: We have just stated that the eigenkets of an observable from a complete set. Let  denote these eigenkets and let  denote an arbitrary ket. Thus

Where ∑ denote a sum over the discrete states and an integration over the continuum states. Since the eigenkets can be assumed to form an orthogonal set we have

     ..........(38)

Thus,  ...............(39)

Since, the above equation holds for an arbitrary ket  , the quantity inside the curely brackets must be a unit operator

Which is usually reffered to the completeness condition. The above equation may be compared with the completeness condition of the Schrodinger wavefunctions

HERMITIAN ADJOINT OF OPERATORS

Here  under operator and transpose change function.

If , Then it is Hermitian.

If A is Hermitian              

If p_x is Hermitian       

This shows px is Hermitian and

Now prove it,

General property:




A is Hermitian.
A is antihermitian.