The Hydrogen Atom - Atomic Structure Chemistry Notes | EduRev

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THE HYDROGEN ATOM

Quantization of Electronic Energy:

Necessity of Repairing Bohr’s theory:

The mathematical framework of the Bohr’s theory was based on the basic assumption of quantization of orbital angular momentum of the electron. It was seen earlier that this theory led to the quantization of electronic energies which formed the basis for explaining the experimental spectra of hydrogen like species such as  H, He+, Li2+, and Be3+. However, this theory was not entirely satisfactory as it failed to provide an interpretation of relative line intensities in the hydrogen spectrum and also failed completely when it was applied to explain the energies and spectra of more complex atoms. In this section, we consider the application of Schrodinger’s wave theory to one-electron atom. In the subsequent section, it will be shown how the principles of this theory can be applied, in a more approximate way, to many electron atoms.

Setting of Schrodinger Equation: 

The time independent form of Schrodinger equation is

The Hydrogen Atom - Atomic Structure Chemistry Notes | EduRev...........(1) Where Hop is the Hamiltonian operator, Etotal is the total nonrelativistic energy and The Hydrogen Atom - Atomic Structure Chemistry Notes | EduRev is the wave function for the total system. Since the hydrogen like system contain two particles, namely, nucleus and electron, it is obvious that wave functionThe Hydrogen Atom - Atomic Structure Chemistry Notes | EduRev depends on the six coordinate variables, three for the electron xe, ye, zand three for the nucleus xn, yn, zn , both sets of coordinates refer to the common origin. The Hamiltonian operator consists of two terms, viz. the kinetic and potential energy terms. The kinetic energy operator will contain two terms, one for the electron and one for the nucleus. Thus, we have

The Hydrogen Atom - Atomic Structure Chemistry Notes | EduRev        ...........(2)
The Hydrogen Atom - Atomic Structure Chemistry Notes | EduRev

All symbols have their usual meanings, Substituting Eq. 1, have

The Hydrogen Atom - Atomic Structure Chemistry Notes | EduRev ..............(3)
The Hydrogen Atom - Atomic Structure Chemistry Notes | EduRev

Equation 3 can be broken into two simpler equations, one involving the free movement of the centre of mass of the atom in space and the other involving the relative motion of the electron with respect to the nucleus within the atom. the two equations are

The Hydrogen Atom - Atomic Structure Chemistry Notes | EduRev............(4)

and 

The Hydrogen Atom - Atomic Structure Chemistry Notes | EduRev.........(5)

with    Etrans + E = Etotal

Equation (4) is simply the Schrodinger equation for a particle of mass (me + mn) , Etrans is the translational kinetics energy associated with the free movement of the centre of mass of the atom through space. Equation 5 is the Schrodinger equation which represents the system in which an electron of reduced mass μ is revilving around the stationary nucleus of positive charge Z at a distance of r. The behavior of this electron can be described by the function ψe and E is the corresponding energy of the electron. The allowed values of electronic energies can be obtained by solving Equation 5 Equation 4 which describes the motion of the centre of mass is of the form as that of the particle in a three dimensional box.

Schrodinger Equation in Terms of Spherical Polar Coordinates:

The solution of Schrodinger equation becomes very much simplify if the equation is expressed in the coordinate system that reflects the symmetry of the system. In the present case the potential energy field is spherical symmetry (V depends only on r), and thus, it is convenient to case the Schrodinger equation 5 into the spherical polar coordinates, θ and φ by using the relations

The Hydrogen Atom - Atomic Structure Chemistry Notes | EduRev

The transformation is length and tedious process, and only the transformed expression is given below:

The Hydrogen Atom - Atomic Structure Chemistry Notes | EduRev

Where ψ is a function of r, θ and φ

Splitting of Schrodinger Equation: Rearranging the above expression, we get

The Hydrogen Atom - Atomic Structure Chemistry Notes | EduRev

Multiplying throughout by r2 and rearranging the resulrant expression, we get

The Hydrogen Atom - Atomic Structure Chemistry Notes | EduRev..........(6)

Since the operator is made up of two terms, one depending on the variable r and the other on the variables θ and φ taken together, we can write the wave function ψ as the product of two functions- one depending on r and the other on θ and φ. Hence, we can write

The Hydrogen Atom - Atomic Structure Chemistry Notes | EduRev............(7)

The function Y is known as spherical harmonics. Substituting Equation 7 in equation 6 we get

The Hydrogen Atom - Atomic Structure Chemistry Notes | EduRev 
The Hydrogen Atom - Atomic Structure Chemistry Notes | EduRev...................(8)

Dividing throughout by Rr Yθ.φ, we get

The Hydrogen Atom - Atomic Structure Chemistry Notes | EduRev
The Hydrogen Atom - Atomic Structure Chemistry Notes | EduRev............(9)

 

Equality shown in equation (9) holds good only when both sides are equal to a constant say, ℓ (ℓ + 1). Thus equation (9) separates into two equations, one depending only one and other on and φ. These are:

Equation involving the variable r

The Hydrogen Atom - Atomic Structure Chemistry Notes | EduRev..........(10)

Equation involving the angles θ and φ

The Hydrogen Atom - Atomic Structure Chemistry Notes | EduRev...............(11)

Multiplying equation (1) by sinθ and rearranging the resultant expression, we get

The Hydrogen Atom - Atomic Structure Chemistry Notes | EduRev..............(12)

Since the operator in equation (11) consists of two terms. One depending on θ and other on φ, we can write the wave function (12) becomes

The Hydrogen Atom - Atomic Structure Chemistry Notes | EduRev

Dividing throughout byThe Hydrogen Atom - Atomic Structure Chemistry Notes | EduRev and rearranging, we get

The Hydrogen Atom - Atomic Structure Chemistry Notes | EduRev..........(13)

The two sides of equation (12) must be equal to a constant, say m2. Thus, we have

The Hydrogen Atom - Atomic Structure Chemistry Notes | EduRev................(14)

And The Hydrogen Atom - Atomic Structure Chemistry Notes | EduRev................(15)

Three Split Expressions of the Schrodinger Equation:

Thus, the Schrodinger equation (6) for the hydrogen like species can be separated into three equations. These are:

  • Equation involving only r

The Hydrogen Atom - Atomic Structure Chemistry Notes | EduRev...............(16)

  • Equation involving only θ

The Hydrogen Atom - Atomic Structure Chemistry Notes | EduRev...............(17)

  • Equation involving only φ

The Hydrogen Atom - Atomic Structure Chemistry Notes | EduRev................(18)

We now consider the acceptable solutions of Equations (16), (17) and (18).

 Solutions of φ Dependent Equation:

Equation (17) has already been solved in connection with the rigid rotator system. Its solution is

The Hydrogen Atom - Atomic Structure Chemistry Notes | EduRev...............(19)

The constant m is called the magnetic quantum number and it represents the quantization of the z-component of the angular momentum since

The Hydrogen Atom - Atomic Structure Chemistry Notes | EduRev
The Hydrogen Atom - Atomic Structure Chemistry Notes | EduRev...............(20)

Thus, the permitted values of z-component of the angular momentum of the electron are given by the expression m(h/2π).

Solutions of θ-Dependent Equation:

Equation (17) has also been solved in the rotator system. The solution is

The Hydrogen Atom - Atomic Structure Chemistry Notes | EduRev .............(21)

WhereThe Hydrogen Atom - Atomic Structure Chemistry Notes | EduRev:Associated legendre polynominals ............(22)

The Hydrogen Atom - Atomic Structure Chemistry Notes | EduRev : legendre polynominals ............(23)

And     The Hydrogen Atom - Atomic Structure Chemistry Notes | EduRev

These solutions are obtained provided the following conditions are satisfied.

The Hydrogen Atom - Atomic Structure Chemistry Notes | EduRev

The quantum number l is called the azimuthal quantum or the subsidiary quantum number and it represents the quantization of the square of total angular momentum according to the equation

 The Hydrogen Atom - Atomic Structure Chemistry Notes | EduRev  ..............(24)

 

Solution of r-Dependent Equation:

With the help of a suitable transformation of independent variable and from the forms of solutions as the variable approaches zero and infinity, it is possible to write in the following more familiar form, known as associated laguerre equation.

The Hydrogen Atom - Atomic Structure Chemistry Notes | EduRev.............(25)

Where the function R is related to the function L through the following transformation scheme.

 The Hydrogen Atom - Atomic Structure Chemistry Notes | EduRev ..............(26)

The terms ρ,j and k are given by

The Hydrogen Atom - Atomic Structure Chemistry Notes | EduRev ..............(27)

The Hydrogen Atom - Atomic Structure Chemistry Notes | EduRev..............(28)

The Hydrogen Atom - Atomic Structure Chemistry Notes | EduRev..............(29)

 

Where The Hydrogen Atom - Atomic Structure Chemistry Notes | EduRev.............(30)

The Hydrogen Atom - Atomic Structure Chemistry Notes | EduRev..............(31)

The solution of Equation (25). as determined by the power series method, is the associated laguerre polynomial of degree (k-j) and order j, and is given by

The Hydrogen Atom - Atomic Structure Chemistry Notes | EduRev...................(32)

Where Lk, the laguerre polynomial of degree k, is given by

The Hydrogen Atom - Atomic Structure Chemistry Notes | EduRev...................(33)

In order that the association laguerre polynomial is the acceptable solution of equation (25) the following quantum restriction has to be satisfied.

k = 1,2,3,................∞...................(34)

Acceptable solutions, as usual, means that the wave function L (ρ) should be a well-behaved function and should vanish as ρ→∞. The later condition that the polynomial must be restricted to a finite number of terms.

This quantum restriction leads to the fact that λ which is equal to K-t(eq. 29) must also be an integer. Let λ be written as n, so that

k = n + 1...................(35 )

Expression of the Normalized R Function:

Thus, the function R as given by Equation (26) is

The Hydrogen Atom - Atomic Structure Chemistry Notes | EduRev...................(36)

Normalizing the above solution through the expression

The Hydrogen Atom - Atomic Structure Chemistry Notes | EduRev

We get   The Hydrogen Atom - Atomic Structure Chemistry Notes | EduRev...................(37)

Where   The Hydrogen Atom - Atomic Structure Chemistry Notes | EduRev...................(38)

And  The Hydrogen Atom - Atomic Structure Chemistry Notes | EduRev...................(39)


Allowed Values of n and its Relation with I:

The constant The Hydrogen Atom - Atomic Structure Chemistry Notes | EduRev is referred to as the principal quantum number. If the associated Laguerre polynomial (Eq32) is not to vanish, we must have

            j≤k                                                                                           …(40) 
j2ℓ + 1 and k = n + ℓ, we have

Since (2ℓ  + 1) ≤ (n + ℓ )

Or        ℓ + 1 ≤ n              ℓ ≤ n - 1...................(41)

Since the allowed values of  The Hydrogen Atom - Atomic Structure Chemistry Notes | EduRev  are 1,2,3,….,it is obvious that the principal quantum number n cannot have a zero value. The allowed values of n are only positive integers. Thus, we have

n = 1,2,3.4,.........
l = 0,1,2,3,.....,(n - 1).................(42)
 

Expressions for the Function Rn.:

The exact form of the function Rn. (r) for the given values of n and The Hydrogen Atom - Atomic Structure Chemistry Notes | EduRev can be determined from Equations (37), (32) and (33). These equations are

The Hydrogen Atom - Atomic Structure Chemistry Notes | EduRev...................(43)
The Hydrogen Atom - Atomic Structure Chemistry Notes | EduRev             ...................(44)

And     The Hydrogen Atom - Atomic Structure Chemistry Notes | EduRev
Where The Hydrogen Atom - Atomic Structure Chemistry Notes | EduRev...................(45)

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