BioPHYSICAL chemistry

where ris the distance between the electron and nucleus, ε 0 is the vacuum
permittivity, −eis the electron charge, and +eis the proton charge. Sub-
stituting this potential into Schrödinger’s equation gives:

(12.2)

The mathematical analysis of this equation is presented in Derivation
box 12.1: the reader may skip to the main text following the box, which
discusses the properties of the general solution.

−∇ +

−

=

Z^222

24 m 0

rr

e

r

(, , ) (, , )θφψ θφ (, , )r

πε

ψθφErψθφ(, , )

CHAPTER 12 THE HYDROGEN ATOM 239

Derivation box 12.1 Solving Schrödinger’s equation for the hydrogen atom

Since the potential energy is determined by the radial distance r, we must substitute the

radial form of the gradient:

(db12.1)

To solve the equation we will use the separation-of-variables approach to produce three

separate equations, each with only one of the three variables. This will lead to three eigenequa-

tions and three quantum numbers, which are called n, l, and ml. Since the potential has only

a radial dependence, we first will consider the radial part separately and define the angular

component as:

(db12.2)

Using this definition we can write the gradient squared as:

(db12.3)

This gives, for the Schrödinger equation (eqn 12.2), the following:

(db12.4)

Separation of variables

Now we shall use the separation-of-variables approach and define:

ψ(r,θ,φ) =R(r)Y(θ,φ) (db12.5)

−+

Z^22

22

2

2

11

mr

rr

rr

r

δψθφ

δ

θφψ θφ

[(,,)]

Λ(,)(,,)) (,,) (,,)

⎧

⎨

⎪

⎩⎪

⎫

⎬

⎪

⎭⎪

+

−

=

e

r

rEr

2

4 πε 0

ψθφ ψθφ

∇= +^2

2

2

11

(, , ) (, , )rr ( (, , ))

rr

rr

r

θφψ θφ

δ

δ

ψθφ 22 Λ^2 (,)(,,)θφψ θφr

Λ^22

2

2

11

(,)

sin sin

θφ sin

θ

δ

δφ θ

δ

δθ

θ

δ

δθ

=+

⎛

⎝

⎜⎜

⎞⎞

⎠

⎟⎟

∇=^2 + + +

2

222

2

2

211 1

sin sin

δ

δ

δ

δθ

δ

rrrr δφ θ

δδ

δθ

θ

δ

δθ

sin

⎡

⎣

⎢

⎢

⎤

⎦

⎥

⎥