F.1 The Feynman Model and Energy Bands
647
We shall require a minimum of two states in order that we can speak of a coupling
between states: let us therefore work with just such a minimum number of states for
simplicity. We consider, therefore, the two equations
i ¯
h
dw 1
dt
= H 11 w 1 + H 12 w 2 ,
(F.1.7)
i ¯
h
dw 2
dt
= H 21 w 1 + H 22 w 2 .
(F.1.8)
We note firstly that if H 12 = H 21 = 0, then states 1 and 2 are not coupled, and the
solutions for them would be simply
w k = C k e
−iH kk t/¯ h
;
k = 1, 2 ,
(F.1.9)
with corresponding probabilities for our system being in state k given by |w k (t)| 2 =
|C k | 2 . Hence, once the system is in state k, it will remain there forever (as the C k
are simply constants in this case).
For the H
+
2 molecular ion, these two solutions correspond pictorially to
+ and
+
, where + represents a bare proton and
represents a hydrogen atom, with its
electron in one of the hydrogenic states [we shall also pare this description down to
the H atom in its 1s state, again for the sake of simplicity]. What is meant by saying
that the two states are uncoupled? Simply that if the electron is at proton 1 at the
start, it will always be with proton 1, and similarly for the electron if it starts out
with proton 2. For the two protons separated by a large distance, this is a realistic
possibility: however, when the proton and H atom are brought closer and closer
together, we know from quantum mechanics that the electron can tunnel through
the potential barrier separating the two protons, so that as the proton and H atom
approach one another, there is an increasing probability that the electron will be
able to pass from the H atom to the bare proton: these two states are then said to be
coupled, as they are no longer entirely independent of one another.
Weak coupling means that it remains meaningful to talk about one or the other
state, because even though the two states influence one another, they still essentially
preserve their separate identities. For H
+
2 we have H 11 = H 22 = E 0 and H 12 =
H 21 = −A, so that our pair of differential equations becomes
i ¯
h
dw 1
dt
= E 0 w 1 − Aw 2 ,
(F.1.10)
i ¯
h
dw 2
dt
= −Aw 1 + E 0 w 2 .
(F.1.11)
Let us try a solution of the form w k = C k e −iEt/¯ h , k = 1, 2, and solve the
resulting pair of algebraic equations for E (and also, in principle, for C 1 and C 2 ;
however, we shall not need them here). We obtain
647
We shall require a minimum of two states in order that we can speak of a coupling
between states: let us therefore work with just such a minimum number of states for
simplicity. We consider, therefore, the two equations
i ¯
h
dw 1
dt
= H 11 w 1 + H 12 w 2 ,
(F.1.7)
i ¯
h
dw 2
dt
= H 21 w 1 + H 22 w 2 .
(F.1.8)
We note firstly that if H 12 = H 21 = 0, then states 1 and 2 are not coupled, and the
solutions for them would be simply
w k = C k e
−iH kk t/¯ h
;
k = 1, 2 ,
(F.1.9)
with corresponding probabilities for our system being in state k given by |w k (t)| 2 =
|C k | 2 . Hence, once the system is in state k, it will remain there forever (as the C k
are simply constants in this case).
For the H
+
2 molecular ion, these two solutions correspond pictorially to
+ and
+
, where + represents a bare proton and
represents a hydrogen atom, with its
electron in one of the hydrogenic states [we shall also pare this description down to
the H atom in its 1s state, again for the sake of simplicity]. What is meant by saying
that the two states are uncoupled? Simply that if the electron is at proton 1 at the
start, it will always be with proton 1, and similarly for the electron if it starts out
with proton 2. For the two protons separated by a large distance, this is a realistic
possibility: however, when the proton and H atom are brought closer and closer
together, we know from quantum mechanics that the electron can tunnel through
the potential barrier separating the two protons, so that as the proton and H atom
approach one another, there is an increasing probability that the electron will be
able to pass from the H atom to the bare proton: these two states are then said to be
coupled, as they are no longer entirely independent of one another.
Weak coupling means that it remains meaningful to talk about one or the other
state, because even though the two states influence one another, they still essentially
preserve their separate identities. For H
+
2 we have H 11 = H 22 = E 0 and H 12 =
H 21 = −A, so that our pair of differential equations becomes
i ¯
h
dw 1
dt
= E 0 w 1 − Aw 2 ,
(F.1.10)
i ¯
h
dw 2
dt
= −Aw 1 + E 0 w 2 .
(F.1.11)
Let us try a solution of the form w k = C k e −iEt/¯ h , k = 1, 2, and solve the
resulting pair of algebraic equations for E (and also, in principle, for C 1 and C 2 ;
however, we shall not need them here). We obtain