ONE-ELECTRON ATOM IN SPHERICAL POLAR COORDINATES

Coulomb Potential Energy:

(1)

1

Time-independent Schrödinger Equation:

(2)

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Transform Laplacian from rectangular to spherical polar coordinates:

2

Use the chain rule,

to find:

(3)

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Assume that the eigenfunction is separable:

(4)

Substituting (3)&(4) into (2) gives:

(5)

** Note that all of the derivatives are now ordinary derivatives so we can replace  by d.

Multiply (5) by

and rearrange terms to obtain:

(6)

Notice that the LHS only involves functions of  while the RHS only involves functions of r and  . Thus, both sides must equal a constant which we'll call C.

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1. FINDING 

Setting the LHS of (6) equal to this constant C gives:

(7)

A solution to (7) is

(8)

The periodicity of  requires that . Now a sin or cos function has this periodicity so let

(9)

which gives:

(10)

This function now has the correct periodicity. Note that the condition that  gives

(11)

Eq. (11) is only satisfied if

m = 0, ±1, ±2,...(12)

Thus, m can only have certain values, i.e. it is a quantum number.

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2. FINDING 

Eq. (6) can now be written as

(13)

Dividing by sin2 and rearranging terms gives:

(14)

Notice that the LHS only involves functions of while the RHS only involves functions of r. Thus, both sides must equal a constant which we'll call (+1).

Setting the LHS of this equation equal to this constant gives an equation for :

(15)

By making a change of variables using z = rcos, Eq. (15) is transformed into a differential equation called the associated Legendre's equation:

(16)

A series solution is found for this equation involving polynomials of z. These polynomials are called associated Legendre functions.

The requirement that  remain finite (i.e. does not blow up) leads to integer values of and certain restrictions on m as follows:

 = 0, 1, 2, 3, ...(17)

and for a given allowed value of ,

m = -, -+1, .., 0, .., -1,  (18)

Thus, we have another quantum number which dictates possible values for the original quantum number m.

3

We can write the associated Legendre functions with the quantum number subscripts as polynomials of z, or to be of more use, as polynomials of cos in the form . The first few polynomials are given below:

00 = 1

10 = cos1±1 = (1-cos2)1/2

20 = 1-3cos22±1 = (1-cos2)1/2cos2±2 = 1-cos2

------The Spherical Harmonic Functions ------

It is customary to multiply the () and () functions to form the so-called spherical harmonic functions which can be written as:

(19)

The first few spherical harmonics are given below:

Y 00= 1

Y 01= cosY ±11= (1-cos2)1/2 e±i

Y 02= 1-3cos2Y ±12= (1-cos2)1/2cos e±iY ±22= (1-cos2) e±i

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3. FINDING R(r) AND THE QUANTIZED ENERGIES

Setting the RHS of Eq. (14) equal to the constant (+1) gives:

(20)

We define two new variables:

(21)

Using these substitutions and multiplying Eq. (20) by R / r2 gives:

(22)

By making a change of variables using r = 2 and dividing by 42, this Eq. (22) becomes:

(23)

As discussed in class, we are led to look for solutions to Eq. (23) of the form

(24)

Substituting this function into Eq. (23) gives:

(25)

Again, a series solution is found involving polynomials. The requirement that F() does not blow up as  demands that  be an integer (which we’ll call n) that obeys:

 = n = +1, +2, +3,...(26)

Note that this third quantum numbern dictates the allowed values of the quantum number  since for a given n, then  can only have values of

 = 0, 1, 2,..., n-1(27)

And remember that the value of a particular  dictates the possible values of m via Eq. (18).

By combining Eqs. (26) with (21), the allowed energies of the atom are found to be:

(28)

This is exactly Bohr's result!!!

By substituting the values for all of the physical constants, we can write this energy in the more convenient form of

(29)

The polynomials that satisfy Eq. (25) depend on  and n which we represent by Fn(). The first few polynomials are shown below:

F10 = 1

F20 = 2-F21 = 

F30 = 6-6+2F31 = 4-2F32 = 2

The radial functions can then be written as shown in Eq. (24), i.e.

Rn(r) = Fn() e-/2

Finally, it is convenient to express  in terms of r, the quantum number n, and the Bohr radius ao. (Recall that ao=0.53 Å.)

 = 2Zr/nao(30)

Thus, the radial functions can be written as

(31)

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SUMMARY

The energy eigenfunction for the state described by the quantum numbers (n,,m) is of the form:

4

Table 7-2 in the book (pg. 243) lists the first ten eigenfunctions. The table is reproduced below.

There are three quantum numbers:

n = 1, 2 ,3,...(Principal quantum no.)

 = 0, 1, 2,..., n-1(Azimuthal quantum no.)

m = -, -+1,..., 0,..., -1,  (Magnetic quantum no.)

The energy of any state only depends on the principal quantum number (for now!) and is given by:

Note that there is usually more than one state with the same energy. These states are called degenerate. The only nondegenerate state is the ground state (1,0,0) with an energy of E1= -Z2(13.6 eV).