Time-Dependent Perturbation Theory in Quantum Mechanics
Delve into time-dependent perturbation theory in quantum mechanics through topics like general treatment to first order, sudden approximation, adiabatic approximation, and resonant phenomena. The lecture explores the treatment of time-dependent perturbations using complete basis sets and perturbation expansion for time-dependent coefficients, providing a comprehensive insight into this complex theoretical framework.
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PHY 741 Quantum Mechanics 12-12:50 AM MWF Olin 103 Plan for Lecture 28: Chap. 18 in Shankar: Time-dependent perturbation theory 1. General treatment to first order 2. Sudden approximation 3. Adiabatic approximation 4. Resonant phenomena 11/06/2017 PHY 741 Fall 2017 -- Lecture 28 1
11/06/2017 PHY 741 Fall 2017 -- Lecture 28 2
Treatment of time-dependent perturbations + t = = ( ) i H t 0 1 ( ) ( ) t H t H H We approach the problem using the complete basis set of = 0 : H 0 0 0 n 0 H n E n It is reasonable to assume that = 0 nt / iE 0 0 ( ) t ( ) t ( ) t c n k e n n n n n 11/06/2017 PHY 741 Fall 2017 -- Lecture 28 3
Treatment of time-dependent perturbations -- continued 0 n = / iE t 0 ( ) t ( ) t k e n n n ( ) = + 0 1 ( ) t i H H t ( ) t dk 0 n = / iE t 1 0 ( ) t k ( ) t 0 i H e n n n dt n Projecting this equation with a particular 0 zero-order state : f ( ) t dk ( ) 0 f 0 n / i E E t f = 0 1 0 ( ) t ( ) t i f H n k e n dt n 11/06/2017 PHY 741 Fall 2017 -- Lecture 28 4
Treatment of time-dependent perturbations -- continued ( ) t t dk ( ) 0 f 0 nt E / i E f = 0 1 0 ( ) t ( ) t i f H n k e n d n Perturbation expansion for time-dependent coefficients ( ) ( ) ( ) ... n t k k t k t k = + + + ^ : 0 n 1 n 2 2 n 0 n dk dt = Zero order equa tion: 0 -order equation for 0: s s s m ( ) 1 i d k dt 0 m 0 n = / i E E t 0 1 0 1 s n ( ) ( ) m H t n k t e n 11/06/2017 PHY 741 Fall 2017 -- Lecture 28 5
Treatment of time-dependent perturbations -- continued = st 0 n 1 -order equation, assuming that k nI 1 m ( ) 1 i dk dt 0 m 0 I / i E E t = 0 1 0 ( ) m H t I e Example: = 1 1 Suppose th a t ( ) ( ) H t H h t 0 for for 0 0 and t t t T where ( ) h t 2sin t T 11/06/2017 PHY 741 Fall 2017 -- Lecture 28 6
Treatment of time-dependent perturbations -- continued For this example 1 m dk dt = 0 for 0 or t t T 0 1 0 m H I ( ) 1 m 2 i dk dt ( ) ( ) + + i t i t = e e mI mI 2 i for 0 t T ( ) ( ) 0 1 0 m H I + + i T i T 2 i ( 1 1 e e mI mI = 1 m ( ) t k + + 2 mI mI ) 0 m 0 where / for E I E t T mI 11/06/2017 PHY 741 Fall 2017 -- Lecture 28 7
For t T ( ) ( ) 0 1 0 m H I + + i T i T 2 i 1 1 e e mI mI = 1 m ( ) t k + + 2 mI mI ( ) 0 m 0 I where / E E mI 4 2 2 1 m 0 1 0 ( ) t k m H I 2 ( ) ( ) ( ( ) ) ( ( ) ) + + + 2 2 sin / 2 sin / 2 T T mI mI + 2 2 + mI mI 4 2 T ( ) 2 ( ) ( ) + + + 0 1 0 m H I mI mI 2 11/06/2017 PHY 741 Fall 2017 -- Lecture 28 8
Treatment of time-dependent perturbations -- continued = 1( ) in the neighbor m T Behavio r of hood of k mI largest T larger T small T xT mI 2 sin T 2 = Note that dx 2 2 x 11/06/2017 PHY 741 Fall 2017 -- Lecture 28 9
Estimating the rate of transitions I f 2 1 I ( ) t k 2 2 f 0 1 0 f H I T ( ) ( ) ( ) + + + 0 f 0 I 0 f 0 I E E E E Fermi Golden rule 0 f E 0 I E 11/06/2017 PHY 741 Fall 2017 -- Lecture 28 10
Example 1 H H atom in presence of electric field representing field as scalar potential eFz e cFp mc i = = 1' representing field as vector potential H z Note that these two are equivalent in the ideal case: = = 2 p 1 i 1 i p m 0 , , z z H z 2 m 0 f 0 I E E 1 i i p m = = 0 0 0 0 0 0 0 , f I f z H I f z I z i 0 0 = f z I fI 11/06/2017 PHY 741 Fall 2017 -- Lecture 28 11
Non-trivial matrix elements: 1/ 2 3 Z a = / Zr a 0 I e 0 3 0 1 2 / 3 Z a Zr a = /2 Zr a 0 cos f e 0 3 0 32 0 11/06/2017 PHY 741 Fall 2017 -- Lecture 28 12
