Download Time-Dependent Schrödinger Equation and Quantum Transitions and more Exercises Quantum Mechanics in PDF only on Docsity!
The Time-Dependent Schrödinger Equation
with applications to
The Interaction of Light and Matter
and
The Selection Rules in Spectroscopy.
Lecture Notes for Chemistry 452/
by
Marcel Nooijen
Department of Chemistry
University of Waterloo
1. The Time-Dependent Schrödinger equation.
An important postulate in quantum mechanics concerns the time-dependence of the wave function. This is governed by the time-dependent Schrödinger equation
i h ∂Ψ∂^ ( , ) x tt^ = H $^ Ψ( , ) x t (1.1)
where H $ is the Hamiltonian operator of the system (the operator corresponding to the classical expression for the energy). This is a first order differential equation in t , which means that if we specify the wave function at an initial time t 0 , the wave function is determined at all later times. Let me emphasize that this means the wavefunction has to be specified for all x at initial time t 0. These initial conditions are familiar from wave equations as discussed in MS Chapter 2. In classical physics we often deal with second order differential equations and in addition the time derivative ∂Ψ ( , ) / x t ∂ t would then need to be specified for all x. Let me emphasize here that although the experimental results that can be predicted from QM are statistical in nature, the Schrödinger equation that determines the wave function as a function of time is completely deterministic.
======= Special solutions: Stationary States (only if H $ is time-independent).
If we assume that the wave function can be written as a product: Ψ( , ) x t =φ ( x ) ( )γ t we
can separate the time dependence from the spatial dependence of the wave function in the usual way. The separation constant is called E and will turn out to be the energy of the system for such solutions
→ ... i h d^^ γ dt^ ( ) t^ = E γ ( ) t → γ ( ) t = e −^ iE t (^ −^ t^0 )/h (1.2)
H^ $^ φ ( x ) = E φ( x ) → H $^ φ n ( x ) = E n φ n ( x ) (1.3)
equation (1.2) is called the time-independent Schrödinger equation and plays a central
role in all of chemistry. Since the operator H $ is Hermitean the eigenfunctions form a complete and (can be chosen to be an ) orthonormal set of functions. Using these
eigenfunctions of H $ special solutions to the time-dependent Schrödinger equation can be expressed as
Ψ ( , ) x t = φ n ( x e ) −^ iE^ n (^ t^ − t^0 )/^ h^ ; Ψ( , x t 0 ) =φ n ( x ) (1.4)
For these special solutions of the Schrödinger equation, all measurable properties are independent of time. For this reason they are called stationary states. For example the probability distribution
Ψ ( , ) x t 2 = Ψ( , x t 0 ) 2 =φ n ( x )^2 , (1.5)
but also
A^ $^ A $^ A $ t^ =^ t 0 ∀ ,^ (1.6) as is easily verified, by substituting the product form of the wave function. Also the probabilities to measure an eigenvalue a (^) k are independent of time, as seen below
A^ $^ ϕ k ( x ) = a k ϕ k ( x ) → P tk ( ) = z−∞ ∞ϕ* k ( x ) Ψ( , ) x t 2 = P tk ( 0 ) (1.7)
The common element in each of these proofs is that the time-dependent phase factor cancels because we have both Ψ( , ) x t and Ψ*^ ( , ) x t in each expression. Let me note that the stationary solutions are determined by the initial condition. If you start off with a stationary state at t 0 , the wave function is a stationary state for all time.
electromagnetic field, and in this way makes a transition to the ground state (eventually). This occurs even if no radiation field is present (spontaneous emission). These are the reasons that stationary states and in particular the ground state is so important.
2. Interaction of Light and Matter.
The total Hamiltonian for a molecule in the presence of monochromatic radiation is given by H = H (^) 0 + H '( ) t , where H 0 is the usual molecular Hamiltonian and H '( ) t represents the interaction with radiation:
H '( ) t = − r^ ⋅ E ( )cos( t ) = − ⋅ E ( ) e i^ t^ + e − i^ t
r (^) r r
μ ω ω^12 μ ω ω ω (2.1)
Very importantly, this Hamiltonian is explicitly time-dependent and this completely changes the picture from the previous section. Later on we will use that the field is not completely
sharp in the frequency and we will assume therefore that the field strength can depend on ω
and have a distribution of frequency components. In eqn. (1) μr is the dipole moment operator,
while
r
E ( ω )is the electric field (a vector) oscillating at angular frequency ω. (For a derivation
of this result see MS problem 13.49). We will want to use the time-dependent Schrödinger Equation. (TDSE). This is a little complicated and we will make the discussion as simple as possible, without losing anything of the essential physics involved. Let us first review the time-dependent SE without radiation, also to establish the notation used in this section. We will be using atomic units throughout.
- Case I, no radiation in a 2-level system (review).
i^ ∂∂Ψ t = H (^) 0 Ψ( ) t (2.2)
Let us assume a 2-level system for simplicity. The time- independent Schrödinger Eqn. has solutions H x E x H x E x
0 1 1 1 0 2 2 2
where we assume known the energies and wave functions. Let us assume that the wave function at time t = 0 is given by Ψ ( t = 0 ) = Φ 1 ( x c ) 1 (^) +Φ 2 ( x c ) 2 (2.4) where c 1 and c 2 are arbitrary coefficients. Then the wave function at time t is given by
Ψ ( ) t = Φ 1 ( x c e ) 1 − iE t^1^ +Φ 2 ( x c e ) 2 − iE t^2 (2.5) Please verify for yourself that this satisfies the TDSE Eqn. (2.2), assuming (2.3). Note that the energy has units radians/s here because we have suppressed h. In this case of no radiation we find that if we would measure the energy of the system we would find E 1 with probability
c e 1 − iE t^12 = c 12 , or E 2 with probability c 22 , independent of time. In particular excited states
would not decay and have infinite lifetimes! Other properties would depend on time however. For long times this picture is clearly deficient (we have not included spontaneous emission in this description which results in finite lifetimes for excited states), but it follows from the QM we taught you sofar.
- Case II. General treatment of radiation in 2-level system. Without loss of generality we can write Ψ ( ) t = Φ 1 ( x c ) 1 ( ) t e − iE t^1^ +Φ 2 ( x c ) (^) 2 ( ) t e − iE t^2 , (2.6) where the coefficients c 1 (^) , c 2 depend on time now. Substituting in the TDSE and using the full Hamiltonian, we should satisfy the eqn:
i^ ∂∂Ψ t − H Ψ = 0 (2.7)
or
(^0 11) 1 1 1 0 1 1 1 1 (^22 2 2 2 0 2 2 2 )
1 1 1 1
2 2 2 2
− − − −
− − − −
i ct x e E c t x e H c t x e H t c t x e
i ct x e E c t x e H c t x e H t c t x e
iE t iE t iE t iE t
iE t iE t iE t iE t
( ) [ ( ) ( ) ( ) ( ) ] '( ) ( ) ( )
( ) [ ( ) ( ) ( ) ( ) ] '( ) ( ) ( )
We note that the terms between square brackets cancel (the reason to write Ψ( ) t as in eqn.
2.6). We will now integrate this Eqn. against Φ 1 ( x )and Φ 2 ( x )respectively. Furthermore we assume (for sake of simplicity) that
z Φ^1 (^^ x^ )^ *^ H^ '^ ( ) t^ Φ^1 (^ x dx )^^ =^ zΦ^2 (^ x^ )^ *^ H^ '( ) t^ Φ^2 (^ x dx )^ =^0 (2.9)
and
i ct V c t Vc t
i ct V c t Vc t
∂ =^ ≡
∂ =^ ≡
(^12 )
(^21 )
and hence we get second order equations ∂ ∂ = − ∂ ∂ = −
(^21) 2 1 (^22) 2 2
c t Vc^ t c t Vc^ t
with known solutions
c 1 ( ) t = cos( Vt + ϕ); c 2 ( ) t = sin( Vt +ϕ ) (2.17)
where ϕ is determined by the initial values of c 1 (^) , c 2. Under these conditions the probabilities
to measure the energy E 1 or E 2 hence oscillate in time as cos (^2 Vt )and sin (^2 Vt )respectively. On average there is equal probability to find E 1 or E 2 , independent of the initial conditions. The oscillation time depends on V , i.e. it is proportional to the strength of the field and to the transition dipole! It does not depend on the energy difference between the two states or the frequency of the applied field.
II-B. Near resonance, short times, weak fields. If we assume that at t = 0 , Ψ ( t = 0 ) =Φ 1 ( x ), i.e. c 1 (^) = 1 ; c 2 = 0 , we can assume that c 1 remains more or less unity and integrate the equation for c 2 (^) ( ) t. This yields
c (^) 2 i (^) 0 V e i^ E^ t^ dt (^) i iV E ei^ E 21
( ) ( ) 21 (^ )^21
( ) (^ )
τ ω (^ )^ ω (^ )
= − τ ω = − ω^ τ
z − − −
The probability to find the energy E 2 upon measurement at time τ would be
c V E e e V E E^
V
E E
V
E
E
V F
2 2 2 i^ E^ i^ E 212 2 212 21
2 212
(^221)
2 2
(^221)
21 2
2 2
( ) ( (^ )) ( )( )
( ) (^ cos(^ ) )^
( ) sin [^ (^ ) ]
( )
sin [ ( ) ] [ ( ) ]
τ ω (^ )^ (^ )
ω ω^ τ
ω ω^ τ
= ω^ τ ω^ τ − −^ − = (^) − − − = (^) − −
− − −
(2.19)
The function F ( ω )defined above as the fraction is famous (meaning you should know what it
looks like), and you can find a sketch of it on page 531 of MS. The function is sharply peaked near resonance and it is the explanation for Bohr's rule that energy is only absorbed if the frequency of the radiation matches the energy difference between two quantum states.
In practice we do not have one precise frequency, but a range or distribution. This can be incorporated by taking an integral over the frequency
c P V
E
E
2 2 21 2 2 d
(^221)
21 2
sin [ ( ) ] [ ( ) ]
∞
z (2.20)
If we use that V ( ω )will vary slowly over the region near resonance (where F ( ω )is large),
and shift variables by calling x = 12 ( ω − E 21 )τ → d ω =^2 τ dx the integral reduces to
P 21 (^) V E (^) (^212) x x dx V E
2 ( ) 2 ( ) (^22221)
τ = τ sin (^ )^ = πτ ( )
−∞
∞
z (2.21)
the rate of transition is obtained as ∂ ∂ =
P 21 V 2 E
τ^2 π^ (^21 )^ (2.22)
The analysis would be completely the same if we would start from state Φ 2 ( x )at t = 0 , and we would find that the rate of the transition to state Φ 1 ( x )is given by ∂ ∂ =
P 12 V 2 E
τ^2 π^ (^21 )^ (2.23)
hence the rate of absorbtion equals the rate of (stimulated) emmission in the presence of radiation. Moreover this rate is proportional to
V^2 ( E 21 ) = [ 12 z Φ 1 ( x ) μ^ r^ Φ 2 ( x dx E E ) ⋅r( 21 )]^2 (2.24)
i.e. proportional to the square of the transition dipole moment and the intensity of the radiation ( E^2 ) at the resonance frequency. The precise formulation would involve a spherical averaging because molecules and hence the transition dipoles are oriented at random.
The stationary state is reached if dN (^) 2 / dt = 0 or N (^) 1 ( ) t = N (^) 2 ( ) t. This is not correct. We are missing something, namely spontaneous emission of radiation that would occur from states Φ 2 even in the absence of resonant radiation. Curiously, this naturally looking phenomenon (finite lifetimes of excited states, even in the absence of radiation) is very hard to derive using QM. The theory that is needed to accomplish a rigorous derivation requires a quantization of the electromagnetic field (i.e. the introduction of photons). It is called quantum field theory. A practical way to account for many of the observed phenomena is to define the process of spontaneous emission. It is an approximation though. We get for the rate of change dN dt B^ E^ N^ t^ N^ t^ A N^ t
2 = ρ( 21 )[ 1 ( ) − 2 ( )] − 2 ( ) (2.28)
At the steady state (equilibrium) we can solve for the intensity of the radiation
ρ( E 21 ) B N ( / AN )
For a black body at temperature T that hypothetically would consists of a two-level system in equilibrium with the radiation it generates and absorbs, we know the population ratio from the Maxwell-Boltzmann distribution law N (^) 1 / N (^) 2 = e − h^^ ν^12 / kT (2.30) The above equation for the radiation density then agrees with the black-body radiation law if
A = 8 c^ π 2 h^ B ν 123 (2.31)
Einstein essentially knew about statistical mechanics, discrete energy levels, and black body radiation and from this he deduced the concepts of spontaneous and stimulated emission and the idea of lasing. He was remarkable, even for a genius.
3. Selection rules in spectroscopy.
A rigorous treatment of electromagnetic radiation (oscillating field) involves the time- dependent Schrödinger equation, usually in the form of time-dependent perturbation theory (see MS and lecture notes in previous section). At a given instant in time the electric field is more or less constant over the region of a (small) molecule, as the wavelength of the radiation (> ~200 nm) is so large compared to molecular dimensions. The relevant quantity that determines intensities in the spectrum is the transition dipole moment between initial and final states
z^ Ψ^ f int^^ μ^ $Ψ^ i^ int d τ^ (3.1)
Here Ψint^ indicates the total internal wave function involving both nuclear and electronic coordinates, but not the rotational part of the wave function. Similarly
μ^ $ α
α α
= (^) ∑ q r^ r^ , (3.2)
the total dipole moment operator involves both nuclei and electrons (indicated through
summation over α ). In order to make the problem managable we distinguish
electrons: normal modes: molecular rotation: R
r
r
r q
i i (3.3)
and the overall internal wavefunction (disregarding rotations) can be written (approximately) as Ψ av int^ = Ψ a (^) ( ; r q r^ )Φ (^) v ( ) q , (3.4) The electronic wave function Ψ a ( ; r q^ r^ )depends on all of the electrons and in addition there is a parametric dependence of the electronic wavefunction on the normal modes (internal coordinates) q. The vibrational part Φ v ( ) q = φ (^) v (^) 1 ( q 1 (^) ) φ v 2 ( q 2 )....is assumed a product of
harmonic oscillator functions for each normal mode qi. If we do not use a subscript we
indicate the whole set of coordinates, e.g. q → l q l qi = q q 1 , 2 ,...q. The rotational wave function
Ω (^) J M , (^) J ( , θ ϕ) determines the probability distribution of the orientation of the molecule in
space. This rotational part is treated differently from the rest. Let us first discuss the problem at a fixed orientation. We can write the overall transition moment in (3.1) as
As shown below this will lead to the selection rule ∆ v = ± 1 , and only excitations of normal modes that lead to a change in dipole moment in the electronic state are allowed (non-zero transition moment)! For example in CO 2 the symmetric stretch cannot be excited (within this approximation), because the dipole moment remains zero. However the two bending modes and the assymetric stretch can be excited. To conclude this analysis we should prove the selection rules ∆ v = ± 1 for pure vibrational transitions in the present (lowest order, harmonic) approximation. In order to prove ∆ v = ± 1 let us consider the one-dimensional integral
z − ∞∞^ ϕ^ w^ ( ) q q^ ϕ^ v ( ) q dq (3.11)
The form of the vibrational wave functions is ϕ v ( ) q = P q ev ( ) −α^ q^2 /^2 , where P qv ( ) is a
polynomial in q of order v. From the orthogonality of the wave functions we then deduce that
z−∞ ∞^ q en^^ −α q^2 /^2 ϕ^ v ( ) q^^ =^0 n^ =^ 0 1, ,...,^ v −^1 (3.12)
Therefore
ϕ (^) w q q ϕ (^) v q dq
w v w v w v w v w v
−∞
∞
z =
R
S|
T|
integral is odd) finite, = -
This proofs the selection rule ∆ v = ± 1 for purely (ro)vibrational transitions.
C. Electronic transitions a ≠ b. We only keep the lead term, and have to realize that Φ w corresponds to the normal modes and equilibrium geometries of the final (excited) state while Φ v corresponds to the corresponding modes in the ground state. Keeping only the first term in Equation (7) we find
μ^ r^ μr
fi =^ ab^0 zΦ^ wfinal ( ) q^ Φ^ vinitial ( ) q dq^ (3.14)
and see that the transition moment depends on the electronic transition dipole at the initial
geometry μr ab^0 and the overlap of the vibrational wave functions in initial and final states. If
this transition moment is zero (often by symmetry) the transition cannot be observed or only very weakly through higher order effects. The vibrational overlaps are the Frank-Condon factors that we discussed qualitatively in class, and which are discussed in MS (but you
should include the phase information in their figures!). The actual calculation is quite involved because the integral does not factorize if the normal modes are different in the two electronic states.
Discussion of rotational selection rules.
The orientation of the transition dipole μr (^) fi depends on the orientation of the molecule in
space, in particular with respect to the electric field. The probability distribution for the
orientation is determined by the rotational wave function Ω J M , J ( , θ ϕ). These functions are
precisely the spherical harmonics, which we have denoted before as Ylm ( , θ ϕ ). To facilitate
the discussion let us consider purely rotational transitions, so that μr (^) fi is just the permanent
dipole of the molecule. A classical dipole would start rotating under influence of an
oscillating electric field. The energy is given by − μ E cos θ, where θ is the angle between
the transition dipole moment and the electric field. In the quantum world this interaction
leads to transitions between rotational eigenstates: Ω J M , J ( , θ ϕ) → Ω I M , I ( , θ ϕ)that are
governed by the by the transition moment
μ^ r θ ϕ θ θ ϕ θ θ ϕ
fi E^ z^ Ω^ I M ,^^ I ( ,^ ) cos^ Ω J M , J ( ,^ )sin d d (3.15)
As discussed in MS, for diatomics these matrix elements are non-vanishing only for ∆ J = ± 1 , ∆ M = 0. These are properties of the spherical harmonics, nothing mysterious.
