522
11 Biomolecular Structure and Interactions
and its eigenvalue will be m ¯
h. Now in order for the wave function of the electron
to return to its original value after a full cycle of the azimuthal angle φ, the value
of m must be an integer. This is an example of a cyclic boundary condition on a
wave function leading to a quantum property. The quantization of the square of
the angular momentum of the electron, L 2 to be l(l + 1) ¯
h 2 , with l a non-negative
integer, is equivalent to a cyclic condition on θ , and also limits the magnitude of
m to be not greater than l. Another quantum number for steady-state waves comes
from the boundary condition that the electron wave must be finite at the proton, and
must drop sufficiently fast at infinity that the total probability of finding the electron
anywhere is one. Energy quantization follows. The result is that the steady-state
electron waves can be labeled by their number of radial nodes, ‘n r ’. The energy of
the system is proportional to the inverse square of the number n ≡ n r + l + 1, called
the ‘principle quantum number in hydrogen’: E n = −13.6 eV/n 2 .
Because the potential energy is a function of the distance of the electron from
the nucleus, but not the angles where the electron might be found, the Schrödinger
equation solutions for bound electrons in a steady state can be expressed as a product
of functions of time, radius, and spherical polar angles, separately. Thus, hydrogenic
electron orbitals take the form
(t, x) = exp (−iE n t/ ¯
h)R nl (r)Y lm (θ, φ) .
(11.18)
The angular functions Y lm are called ‘spherical harmonics’. They separate into
functions of θ and φ as Y lm (θ, φ) = N Y P l
m (θ ) exp (imφ), where the constant N Y is
selected to make
((Y lm ) ∗ Y lm d(cos θ)dφ = 1.
For an isolated hydrogen atom, its energy and vector angular momentum are
conserved. Moreover, the energy
H , the square of angular momentum L 2 , and
one component of the angular momentum (taken as
L z ) can be measured without
disturbing the others. (The operators for these three observables commute.) The
electron spin along one axis is also quantized to have values S z = m s ¯
h with
m z = −1/2 or 1/2. So, we can label the hydrogenic waves by the quantum
numbers associated with the quantization of the energy, the magnitude of the angular
momentum, the z-component of the angular momentum, and the z-component of the
electron spin: n, l, m, m s . When there are a number of possible quantum states for a
given energy, the set of states is called ‘degenerate’. Sets of quantum states separated
from others by a relatively large energy gap are said to be within one ‘shell’. For
hydrogen, the number of allowed states for the electron in a given shell labeled by
‘n’ add up to 2
n−1
0 (2l + 1) = 2n 2 . Transitions from one shell to another, from a
bound state to an unbound state, or from an unbound state to a bound one, requires
an energy exchange with an external agent, which can be a photon or a nearby atom.
Historically, transitions between hydrogenic atomic states in alkaline metals
were seen in spectroscopes and labeled with the letters s, p, d, and f from the
appearance on film of the corresponding spectral lines, being: sharp, principal,
diffuse, and fundamental. Later, these observations were found associated with
electron transitions from states with l = 0, 1, 2, 3 to other states. We still use
the letters s, p, d, f instead of the quantum numbers 0, 1, 2, 3 to label the angular
11 Biomolecular Structure and Interactions
and its eigenvalue will be m ¯
h. Now in order for the wave function of the electron
to return to its original value after a full cycle of the azimuthal angle φ, the value
of m must be an integer. This is an example of a cyclic boundary condition on a
wave function leading to a quantum property. The quantization of the square of
the angular momentum of the electron, L 2 to be l(l + 1) ¯
h 2 , with l a non-negative
integer, is equivalent to a cyclic condition on θ , and also limits the magnitude of
m to be not greater than l. Another quantum number for steady-state waves comes
from the boundary condition that the electron wave must be finite at the proton, and
must drop sufficiently fast at infinity that the total probability of finding the electron
anywhere is one. Energy quantization follows. The result is that the steady-state
electron waves can be labeled by their number of radial nodes, ‘n r ’. The energy of
the system is proportional to the inverse square of the number n ≡ n r + l + 1, called
the ‘principle quantum number in hydrogen’: E n = −13.6 eV/n 2 .
Because the potential energy is a function of the distance of the electron from
the nucleus, but not the angles where the electron might be found, the Schrödinger
equation solutions for bound electrons in a steady state can be expressed as a product
of functions of time, radius, and spherical polar angles, separately. Thus, hydrogenic
electron orbitals take the form
(t, x) = exp (−iE n t/ ¯
h)R nl (r)Y lm (θ, φ) .
(11.18)
The angular functions Y lm are called ‘spherical harmonics’. They separate into
functions of θ and φ as Y lm (θ, φ) = N Y P l
m (θ ) exp (imφ), where the constant N Y is
selected to make
((Y lm ) ∗ Y lm d(cos θ)dφ = 1.
For an isolated hydrogen atom, its energy and vector angular momentum are
conserved. Moreover, the energy
H , the square of angular momentum L 2 , and
one component of the angular momentum (taken as
L z ) can be measured without
disturbing the others. (The operators for these three observables commute.) The
electron spin along one axis is also quantized to have values S z = m s ¯
h with
m z = −1/2 or 1/2. So, we can label the hydrogenic waves by the quantum
numbers associated with the quantization of the energy, the magnitude of the angular
momentum, the z-component of the angular momentum, and the z-component of the
electron spin: n, l, m, m s . When there are a number of possible quantum states for a
given energy, the set of states is called ‘degenerate’. Sets of quantum states separated
from others by a relatively large energy gap are said to be within one ‘shell’. For
hydrogen, the number of allowed states for the electron in a given shell labeled by
‘n’ add up to 2
n−1
0 (2l + 1) = 2n 2 . Transitions from one shell to another, from a
bound state to an unbound state, or from an unbound state to a bound one, requires
an energy exchange with an external agent, which can be a photon or a nearby atom.
Historically, transitions between hydrogenic atomic states in alkaline metals
were seen in spectroscopes and labeled with the letters s, p, d, and f from the
appearance on film of the corresponding spectral lines, being: sharp, principal,
diffuse, and fundamental. Later, these observations were found associated with
electron transitions from states with l = 0, 1, 2, 3 to other states. We still use
the letters s, p, d, f instead of the quantum numbers 0, 1, 2, 3 to label the angular