We found that R(r) = e^-cr r^l M(r) with M(r) = S b_j r^j

But we had to truncate M after term j = k, where k = N - l - 1

So M(r) = S b_j r^j

We also found that c = Z/(na) & a = h²/[m (e’)²]

R_nl(r) = r^l e^-[Zr/(na)] S b_j r^j

and y _nlm (r, q , f ) = R_nl(r) Y_l^m (q , f )

= R_nl(r) S_lm(q ) e ^imf Ö (2p )

Nodes of R_nl:

R_nl = 0 r = ¥

= 0 r = 0, l not equal to 0

= 0 M(r) = 0

M(r) is a polynomial of degree (highest power of r) n-l-1

So there are (n-l-1) roots or nodes of this polynomial

Typical energy units:

1 eV (electron volt) = 1.6022 x 10^-19 J = 1.6022 x10^-12 erg

Find the Bohr radius (radius of electron in first Bohr orbit, a_o) :

For the ground state of the H atom, E = - 13.60 eV

(This is the ionization energy)

Reduced mass = m _H = m_p m_e/(m_p + m_e),

where m_p = mass of proton, m_e = mass of electron;

m_p + m_e » m_p so m _H = m_e

Then a = h²/[m (e’)²] Þ a_o = h²/[ m_e (e’)²] = 0.529 Angstroms

(But Bohr considered the electron to be confined to a circle. This is impossible, according to the Uncertainty Principle.)

Hydrogen-Like Atom Wavefunctions:

y _nlm (r,q ,f ) = R_nl (r) Y_l^m (q ,f ) = R_nl (r) S_l,m (q ) e^imf /Ö (2p )

See Table 5.1 for the values of S_l,m (q ), Table 6.1 for R_nl (r)

Ground State Wavefunction: n=1, l=0, m=0

Y₀⁰ (q ,f ) = S_0,0 (q ) e⁰/Ö (2p ) = (Ö 2/2) × 1 × /Ö (2p ) = 1/Ö (4p )

R₁₀ (r) = b₀ e^-Zr/a

Find b₀ by normalization:

ò _0¥ ú R₁₀ (r)ú ² r² dr = 1 = ú b_0ú ^{2 ò} _0¥ e^-2Zr/a r² dr

= ú b_0ú ² 2!/(2Z/a)³ Þ b₀ = (2Z/a)^3/2/Ö 2 = 2 (Z/a)^3/2

Therefore,

R₁₀ (r) = 2 (Z/a)^3/2 e^-Zr/a

And

y ₁₀₀ (r,q , f ) = 2 (Z/a)^3/2 e^-Zr/aÖ (4p ) = (Z/a)^3/2 e^{-Zr/aÖ p}

Old spectroscopic notation relates the values of l to letters:

l = 0(s), 1(p), 2(d), 3(f), 4(g), 5(h), 6(i), 7(k).... (no j)

For n=2, l = 0,1. For l = 0, m = 0 (y _2s);

For l = 1, m= -1 (y _2p-1 or y _21-1), 0 (y _2p0 or y ₂₁₀), 1 (y _2p+1 or y ₂₁₁)

Using Tables 5.1 & 6.1,

y _21-1 (r,q , f ) = R_2p (r)Y₁^-1 (q ,f )

= 1/(2Ö 6) (Z/a)^5/2 r e^-Zr/2a 1/Ö (2p ) Ö 3/2 sin q e^-if

= 1/(8Ö p ) (Z/a)^5/2 r e^-Zr/2a sin q e^-if

The probability of finding the electron between r+dr, q +dq , and

f +df is

½ y ÷ ² dt = ½ R_nl (r)÷ ² r² dr ½ Y_l^m (q ,f )÷ ² sin q dq df ,

where dt = r² dr sin q dq df . If we just consider the probability of finding the electron in the shell r+dr with no restrictions on q & f , then we can integrate over q & f to get the Radial Distribution Function (i.e. the probability density for the radial part of the wavefunction):

ò ₀^2p df ò _0p sin q dq ½ Y_l^m (q ,f )÷ ^{2 ½} R_nl (r)÷ ² r² dr = ½ R_nl (r)÷ ² r² dr

since the Spherical Harmonics are normalized. The Radial Distribution Functions for n = 1, 2, 3 are plotted in Fig. 6.8

The function a [R_nl (r)÷ ² r² dr is plotted in Fig. 6.9. Note that the maximum in the 1s radial distribution function, a[R₁₀(r)]²r², occurs at r = a. How could you prove this mathematically? (Find the value of r for which ¶ /¶ r[a[R₁₀(r)]²r²] = 0.)

Since y _nlm goes as e^imf , the hydrogenlike y ‘s are imaginary. For convenience, it is easier to use real wavefunctions. We can construct the real wavefunctions by taking certain linear combinations of the imaginary wavefunctions. They will still be an eigenfunction of the Hamiltonian operator with the same eigenvalue. We already proved that

If H y _N = E y _N, N = 1,...,m

then H (y ₁ + ... + _{y m}) = E (y ₁ + ... + _{y m}).

In other words, a linear combination of eigenfunctions of an operator will also be an eigenfunction of the operator with the same eigenvalue.

From Tables 5.1 & 6.1

y _2p1 = 1/(8Ö p ) (Z/a)^5/2 r e^-Zr/(2a) sin q e^if

y _2p-1 = 1/(8Ö p ) (Z/a)^5/2 r e^-Zr/(2a) sin q e^-if

2cos f = e^if + e^-if

So y _2p1 + y _2p-1 = 1/(8Ö p ) (Z/a)^5/2 r e^-Zr/(2a) sin q (e^if + e^-if )

= 1/(8Ö p ) (Z/a)^5/2 r e^-Zr/(2a) sin q 2cos f

= 1/(4Ö p ) (Z/a)^5/2 r e^-Zr/(2a) sin q cos f

Remember x = r sin q cos f . So y _2p1 + y _2p-1

= 1/(4Ö p ) (Z/a)^5/2 e^-Zr/(2a) x

Define y _2px = 1/Ö 2 (y _2p1 + y _2p-1)

= 1/[4Ö (2p )] (Z/a)^5/2 e^-Zr/(2a) x,

where 1/Ö 2 is a normalization factor. Prove that y _2px is normalized. In the next chapter, we will see that the Spherical Harmonics are orthonormal or orthogonal functions. That is

ò Y_l^m Y_l’^m’ dt = d _{ll’d mm’}, where d _ll’=0 unless l=l’

So if l=1 & l’=-1, the integral is zero.

ò ú y _2pxú ² dt = ò y _2px^* y _2px dt

= ò 1/Ö 2 (y _2p1 + y _2p-1)^* 1/Ö 2 (y _2p1 + y _2p-1) dt

= 1/2 (cos 4p + i× 0 - cos 0 - i× 0) = 1/2 (1-1) = 0

So

ò ú y _2pxú ² dt = 1/2 ( 1 + 0 + 0 + 1) = 1

Other real wavefunctions for n=2:

y _2py = 1/(iÖ 2) (y _2p1 - y _2p-1)

= 1/(4Ö (2p )) (Z/a)^5/2 r sin q sin f e ^-Zr/(2a)

= 1/(4Ö (2p )) (Z/a)^5/2 y e ^-Zr/(2a)

y _2pz = y _2p0 = 1/Ö p (Z/a)^5/2 z e ^-Zr/(2a)

Table 6.2 gives the real wavefunctions for n=1,2,3

Are the real wavefunctions eigenfunctions of L² & L_z?

L^{2 y} _2p-1 = l(l+1)h^2y _2p-1 = 2 h^2y _2p-1

L^{2 y} _2p1 = l(l+1)h^2y _2p1 = 2 h^2y _2p1

L² (y _2p1 + y _2p-1) = 2 h²(y _2p1 + y _2p-1) yes

But L_{zy 2p-1} = m hy _2p-1 = - hy _2p-1

L_{zy 2p1} = m hy _2p1 = hy _2p1

L_z(y _2p1 + y _2p-1) = hy _2p1 - hy _2p-1= h (y _2p1 - y _2p-1) no

Fig. 6.14 shows the probability densities for some hydrogen atom states

Fig. 6.13 shows the shapes of some hydrogen atom orbitals. The 2s orbital has one spherical node (not visible); the 3s orbital has two spherical nodes (not visible). The 3p_z orbital has a spherical node (dashed line) & a nodal plane (xy plane). The 3d_z**2 has two nodal cones. The 3d_x**2-y**2 orbital has two nodal planes.

When the hydrogen atom is put into an external magnetic field, a triplet of lines is observed for the 1s ® 2p transition, rather than the single line seen in the absence of a magnetic field. This means that, in the presence of the external magnetic field, the 2p state is no longer degenerate.

Magenetic Dipole: The motion of an electron around a closed loop results in a magnetic dipole vector m of magnitude

÷ m ½ = m = iA

where i is the current (1 amp = 1 Coul/sec) & A is the area of the loop in square meters. If the loop is circular,

A = p r²

where r is the radius. The current, i, is equal to

i = Qv/(2p r)

where Q is the charge & v is the velocity. So

m = iA = [Qv/(2p r)] p r² = Qvr/2.

For a noncircular loop, the magnetic moment due to orbital motion is

m _L = Q/2(r x v).

Since L = r x p & p = mv, then

m _L = Q/(2m) L

where the direction of m & L is perpendicular to the plane of motion.

For an electon, m _L = -e/(2m_e) L,

where -e is the charge on the electron & m_e is the mass of the electron.

Since ½ L½ = hÖ [l(l+1)],

then ½ m _L½ = ½ -e/(2m_e) L½ = ½ -e½ /(2m_e) ½ L½

= e/(2m_e) hÖ [l(l+1)].

The Bohr magneton is defined by

b _e = eh/(2m _e) = 9.274x10^-24 J/T

T=Tesla; 1 Tesla = 1 NC^-1 m^-1 s = 1 N/(amp× m)

So ½ m _L½ = b _e Ö [l(l+1)].

The energy of interaction between a magnetic dipole, m , & an external magnetic field is

E_B = -m × B = e/(2m) L × B.

Assume the magnetic field is applied in the z-direction:

B = B k

and E_B = e/(2m) (L_xi + L_yj + L_zk) × (Bk) = e/(2m) L_zB

E_B = (b _e/h) L_zB

So the corresponding Hamiltonian operator for the interaction of the electron with the magnetic field is

H_B = (b _e/h) L_zB.

So the Hamiltonian that describes the behavior of the hydrogen atom in the magnetic field is the sum of the Hamiltonian in the absence of the field & H_B:

(H + H_B) y = E y .

Since we know the eigenfunctions of L_z, we know y :

y = R(r) Y_l^m(q , f ).

(H + H_B) y = (H + H_B) R(r) Y_l^m(q , f )

= H R(r) Y_l^m(q , f ) + H_B R(r) Y_l^m(q , f )

H R(r) Y_l^m(q , f ) = (-Z²/n²) [(e’) ²/(2a)] R(r) Y_l^m(q , f )

H_B R(r) Y_l^m(q , f ) = (b _e/h) L_zB R(r) Y_l^m(q , f )

= (b _e/h) B R(r) L_z Y_l^m(q , f )

= (b _e/h) B R(r) mhY_l^m(q , f )

= b _em B R(r)Y_l^m(q , f )

(H + H_B) y = {(-Z²/n²) [(e’) ²/(2a)] + b _em B} R(r)Y_l^m(q , f )

So E = {(-Z²/n²) [(e’) ²/(2a)] + b _em B}

For each n, there is a different energy depending on m. This removes the m-fold degeneracy.

In the presence of the magnetic field, the n=2 level (1 line)

E_2s = E_2p = E₂ = (-Z²/4) [(e’) ²/(2a)]

is split into 3 levels (3 lines) with

E_2p+1 = E₂ + b _eB (m = +1)

E_2p0 = E_2s = E₂ (m = 0)

E_2p-1 = E₂ - b _eB (m = -1)