1. Quantum Two 1

2. 2

3. Time-Dependent Perturbations 3

4. Pulsed Perturbations 4

5. In the last segment we derived a basic result of time-dependent perturbation theory, namely an expression for the probability of a transition from the n th to the m th unperturbed eigenstate of, induced by weak time-dependent perturbation V(t) between time t 0 and t, correct to lowest non-trivial order in the perturbation. 5

6. In the last segment we derived a basic result of time-dependent perturbation theory, namely an expression for the probability of a transition from the n th to the m th unperturbed eigenstate of, induced by weak time-dependent perturbation V(t) between time t 0 and t, correct to lowest non-trivial order in the perturbation. 6

7. In this segment, we apply this formula to a so-called pulsed perturbation that vanishes at times in the past, and times in the future, but is non-zero over some time interval, during which it induces transitions in the system of interest. 7

8. In such a situation we can extend the integration from minus infinity to plus infinity to obtain the total probability that a transition is induced by the pulse from energy level n to energy level m, i.e., where is simply the Fourier component of the perturbing matrix element connecting the two states involved in the transition, evaluated at a frequency corresponding to the energy difference between the two states involved. 8

9. In such a situation we can extend the integration from minus infinity to plus infinity to obtain the total probability that a transition is induced by the pulse from energy level n to energy level m, i.e., where is simply the Fourier component of the perturbing matrix element connecting the two states involved in the transition, evaluated at a frequency corresponding to the energy difference between the two states involved. 9

10. In such a situation we can extend the integration from minus infinity to plus infinity to obtain the total probability that a transition is induced by the pulse from energy level n to energy level m, i.e., where is simply the Fourier component of the perturbing matrix element connecting the two states involved in the transition, evaluated at a frequency corresponding to the energy difference between the two states involved. 10

11. In such a situation we can extend the integration from minus infinity to plus infinity to obtain the total probability that a transition is induced by the pulse from energy level n to energy level m, i.e., where is simply the Fourier component of the perturbing matrix element connecting the two states involved in the transition, evaluated at a frequency corresponding to the energy difference between the two states involved. 11

12. In such a situation we can extend the integration from minus infinity to plus infinity to obtain the total probability that a transition is induced by the pulse from energy level n to energy level m, i.e., where is simply the Fourier component of the perturbing matrix element connecting the two states involved in the transition, evaluated at a frequency corresponding to the energy difference between the two states involved. 12

13. As an example, we consider a 1D harmonic oscillator 13

14. As an example, we consider a 1D harmonic oscillator, subjected to a pulsed, but spatially uniform electric field 14

15. Thus, our unperturbed Hamiltonian is Let us assume that the oscillator is initially (at t =  ∞ ) in its ground state, when a perturbing electric field pulse is applied corresponding to a potential of the form In this expression, represents the spatially uniform, but time-dependent force exerted by the field on the charged harmonically-bound particle. 15

16. Thus, our unperturbed Hamiltonian is Let us assume that the oscillator is initially (at t =  ∞ ) in its ground state, when a perturbing electric field pulse is applied corresponding to a potential of the form In this expression, represents the spatially uniform, but time-dependent force exerted by the field on the charged harmonically-bound particle. 16

17. Thus, our unperturbed Hamiltonian is Let us assume that the oscillator is initially (at t =  ∞ ) in its ground state, when a perturbing electric field pulse is applied corresponding to a potential of the form In this expression, represents the spatially uniform, but time-dependent force exerted by the field on the charged harmonically-bound particle of charge e. 17

18. For convenience we consider a Gaussian pulse envelope that peaks at t = 0. Our goal is to find the probability that the particle is left by this pulse in the n th excited state. Provided the pulse strength is sufficiently low, the transition probability can then be written where... 18

19. For convenience we consider a Gaussian pulse envelope that peaks at t = 0. Our goal is to find the probability that the particle is left by this pulse in the n th excited state. Provided the pulse strength is sufficiently low, the transition probability can then be written where... 19

20. For convenience we consider a Gaussian pulse envelope that peaks at t = 0. Our goal is to find the probability that the particle is left by this pulse in the n th excited state. Provided the pulse strength is sufficiently low, the total transition probability can then be written where... 20

21. For convenience we consider a Gaussian pulse envelope that peaks at t = 0. Our goal is to find the probability that the particle is left by this pulse in the n th excited state. Provided the pulse strength is sufficiently low, the total transition probability can then be written where... 21

22. in which is the matrix element of the position operator between the ground state and the n th excited state. Clearly, the first-order transition amplitude vanishes except for the first excited state, i.e., n = 1. 22

23. in which is the matrix element of the position operator between the ground state and the n th excited state. Clearly, the first-order transition amplitude vanishes except for the first excited state, i.e., n = 1. 23

24. the matrix element of the position operator between the ground state and the n th excited state is Clearly, the first-order transition amplitude vanishes except for the first excited state, i.e., n = 1. 24

25. the matrix element of the position operator between the ground state and the n th excited state is Clearly, the first-order transition amplitude vanishes except for the first excited state, i.e., n = 1. 25

26. For the Gaussian pulse, evaluation of the Fourier integral is straightforward: Combining these results, we find that, long after the pulse has passed through, the probability for the oscillator to have been excited to the n = 1 state is 26

27. For the Gaussian pulse, evaluation of the Fourier integral is straightforward: Combining these results, we find that, long after the pulse has passed through, the probability for the oscillator to have been excited to the n = 1 state is 27

28. For the Gaussian pulse, evaluation of the Fourier integral is straightforward: Combining these results, we find that, long after the pulse has passed through, the probability for the oscillator to have been excited to the n = 1 state is 28

29. For the Gaussian pulse, evaluation of the Fourier integral is straightforward: Combining these results, we find that, long after the pulse has passed through, the probability for the oscillator to have been excited to the n = 1 state is 29

30. Note the transition probability goes to zero as the width τ of the pulse goes to zero. It also goes to zero at large τ due to the Gaussian envelope, and has a maximum as a function of τ at 30

31. Note the transition probability goes to zero quadratically as the width τ of the pulse goes to zero. It also goes to zero at large τ due to the Gaussian envelope, and has a maximum as a function of τ at 31

32. Note the transition probability goes to zero quadratically as the width τ of the pulse goes to zero. It also goes to zero at large τ due to the Gaussian envelope, and has a maximum as a function of τ at 32

33. Note the transition probability goes to zero quadratically as the width τ of the pulse goes to zero. It also goes to zero at large τ due to the Gaussian envelope, and has a maximum as a function of τ at 33

34. 34