Atomic Transitions and Selection Rules

Transcription

1 Atomic Transitions and Selection Rules The time-dependent Schrodinger equation for an electron in an atom with a time-independent potential energy is ~ 2 2m r2 (~r, t )+V(~r ) (~r, We found that the solutions to this equation can be found via separation of variables and superposition: (~r, t )= X n c n n (~r )exp( ie n t/~) (n is shorthand for the set of quantum numbers that defines an eigenstate of the Hamiltonian)

2 Clicker Question (~r, t )= X n c n n (~r )exp( ie n t/~) What is the probability that an energy E i is measured? 1. c n 2. c i 3. c i 2 4. c i ψ i 2 5. c i E i 2

3 Clicker Question Suppose the energy is measured and the result is E 3. We now know that the quantum state is ψ 3, and that c n =0 for all values of n not equal to 3. If we wait a time t and then measure the energy again, what is the probability we will measure a value E n (for n not equal to 3)? 1. Zero! 2. Roughly zero for small values of t, but it increases with time. 3. Non zero and a constant value, regardless of the value of t.

4 Time-Dependent Potential Energy Now suppose the potential energy is given by V (~r, t )=V 0 (~r )+V 1 (~r, t ), where V 0 is the original potential energy associated with the atom and V 1 is some time-dependent perturbation (such as due to incident electromagnetic radiation). Now the solution to the time dependent Schrodinger equation will change, and since V 1 is time dependent, separation of variables will no longer work. If V 1 <<V 0, we call V 1 a perturbation. The idea is it still makes sense to think of the electron as being a superposition of eigenstates of V 0, but now the coefficients are time-dependent. (~r, t )= X n c n (t) n (~r )exp( ie n t/~) This means that transitions can occur.

5 Transitions and Selection Rules (~r, t )= X n c n (t) n (~r )exp( ie n t/~) Solving the time-dependent Schrodinger equation to determine c n (t) is well outside the scope of this class (take Atomic and Molecular Physics). It turns out that if the electron is initially in state i (c i (t=0) = 1 and c other (t=0) = 0), then the probability per unit time that a transition to state f occurs is given by where i!f / <f V 1 (~r Z, t ) i > 2, <f V 1 (~r, t ) i >= f V 1 i d 3 r For the case where V 1 is due to unpolarized radiation i!f = Z ~ 2 f~r i d 3 r (! if )

6 Example: 2s to 1s transition Consider the transition of hydrogen from 2s to 1s due to radiation polarized in the z-direction: Z 2 2!1 / 1z 2 d 3 r Z 100z 200 d 3 / Z 0 cos sin d Is this integral zero or non-zero? 1. Zero 2. Non-Zero

7 Selection Rules The selection rules for dipole transitions result from determining when the overlap integral is not zero: Z f ~r i d 3 r 6= 0 For hydrogen-like atoms (one electron), the resulting selection rules are: l = ±1 m l =0, ±1 For multi-electron atoms: J =0, ±1 M J =0, ±1 (J=0 to 0 not allowed) For ideal LS coupling S =0 M S =0 This implies that For 1-e transitions, L =0, ±1 l = ±1

8 Applications to Helium Due to ΔS=0, triplet states can t transition to singlet states. Metastable states: states that can t transition via the dipole transition. Forbidden Transitions. Transitions via magnetic dipole or electric quadrapole interactions.

9 Semi-Classical Model of Transitions Some insight to the nature of transitions can be obtained from a semi-classical model. Classically, we know that radiation is emitted when charged particles accelerate (due to a changing electric field inducing a magnetic field). The electric field can often be approximated by using the dipole moment: ~p = X i q i ~r i E ~ 1 3(~p ˆr)ˆr ~p 4 0 r 3 The Larmor formula gives the power of emitted radiation due to an accelerating charge: P = q2 a c 3

10 Semi-Classical Model of Dipole Transitions A harmonically oscillating dipole can be expressed as ~p = ~p 0 e iwt After expressing the acceleration in terms of the dipole moment, the Larmor formula P = µ 0! 4 p c Now, for quantum mechanics, the dipole moment is due to the electron cloud. ~p = e Z ~r (~r ) 2 d 3 r

11 Z Clicker Question ~p = e ~r (~r ) 2 d 3 r Is the dipole moment zero or non-zero if the electron is in a stationary state (with no time-dependent potential energy)? 1. zero, for any quantum state 2. zero for some, non-zero for other quantum states 3. non-zero for all quantum states

12 Z Clicker Question ~p = e ~r (~r ) 2 d 3 r Is the dipole moment zero or non-zero if the electron is in a stationary state (with no time-dependent potential energy)? 1. zero, for any quantum state 2. zero for some, non-zero for other quantum states 3. non-zero for all quantum states Thus stationary states don t emit any radiation, and again we see transitions don t occur if there is no time-dependent potential energy.

13 Semi-Classical Model of Dipole Transitions Now, due to the time-dependent potential energy, let s let the electron have a foot in both the initial and final states: (~r, t )=c i (t) i (~r )exp( ie i t/~)+c f (t) f exp( ie f t/~) ~p = e Inserting, we find where Z ~r (~r ) 2 d 3 r ~p = c i c f ~p 0 exp(i!t)+c i c f ~p 0 exp( i!t) ~p =2Re[c i c f ~p 0 exp(i!t)] Z ~p 0 = e i ~r f dv ~! = E i E f p 0 is the overlap integral. Determining when this integral is not equal to zero gives us the selection rules.

14 Spontaneous Emission Why does an excited atom, out in space, far from any source of electromagnetic radiation, de-excite?

15 GRE QUESTIONS

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17 X-Ray Line Spectra Transitions of valence electrons result in UV, visible, infrared, and radio emission lines. Multi-electron atoms with a large value of Z can emit X-ray emission lines: If an electron from an inner shell is ejected (say due to a collision with a high-energy electron or photon), an electron from a higher energy state can jump into the hole. A cascade occurs due to the holes left behind by the jumping electrons.

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21 generalizing (for other transitions): f = k 1 (Z k 2 )