Wittig's approach to Landau-Zener made even simpler
A. I. Chichinin a,b)
a)
Institute of Chemical Kinetics and Combustion, Institutskaya,3, Novosibirsk, 630090 Russia
b)
Novosibirsk State University, Pirogova Str.,2, Novosibirsk, 630090 Russia
The Landau-Zener formula was derived in 1932 and since then has remained an important
tool in molecular dynamics and spectroscopy. In 2005, C. Witting proposed a simple
derivation of the formula,[1] it looks very useful and probably will be widely used. In this
note a new derivation of Landau-Zener formula is proposed.[2] Several shortcomings of the
Witting's derivation are overcome; now it is really "a single line" derivation.
Let there be two nonperturbed diabatic states, labeled 1 and 2, and the energy difference
between them is E12 (t )  E1 (t )  E2 (t )   t  v( F1  F2 )t , where v is velocity, t is time, and
F1  dU1 / dr and F2  dU 2 / dr are the slopes of the potential energy curves. We search for
the perturbed wave function in the form A(t )1 (r , t )  B(t ) 2 (r , t ) , where 1 (r , t ) and
 2 (r , t ) are eigen functions of the nonperturbed Hamiltonian. The perturbation is given by
time-independent off-diagonal matrix elements H12 , and | H12 | 12 . Putting the wave
function into the time-dependent Schrödinger equation yields the differential equation
B(t )  i tB(t )  122 B(t )  0 ,
(1)
where B is a complex function, and all other parameters are real. We assume that | B() | 1
and the aim is to find | B () | B f . After dividing (1) by tB and integrating over t one
obtains:


ln B f   ( B / B) dt  (1/ i )  (122  B / B) / t  dt  I x / i ,.


(2)
here the integral at the right side is denoted as I x . From Eq. (1) it follows that the expression
in square brackets has no pole at t  0 . The I x integral may
be calculated from the relation I x  IC  0 , where I C is the
contour integral, see figure. The I C integral may be obtained
IC
t
from integral of Eq. (2) by replacing real t by complex
Ix
i
variable z | z | e ( | z |  ):
IC 
 
2
12

 B( z ) / B( z )  / z dz  i  (122  B / B)| z| d  i122 ,
0
(3)
where the upper and the lower sign correspond to semicircle in upper and lower part of the
complex plane, respectively. According to Eq. (1), ( B / B)|z| ~ z 2 , hence this term
vanishes. Substituting the result from Eq. (3) in Eq. (2) gives B f  exp(122 /  ) . Since B
is a wave function coefficient, it should be | B | 1 ; this condition makes one of the signs
impossible. Finally, we obtain the Landau-Zener formula for the probability of nonadiabatic
transition: PLZ | B f |2  exp(2122 / |  |)  exp(2 2122 / v | F1  F2 |) .
1. C. Wittig, J. Phys. Chem. B, 109, 8428 (2005).
2. A. I. Chichinin, J. Phys. Chem. B, 117, 6018 (2013).