1. Quantum Two

3. Time Dependent Perturbations

4. Time Dependent Perturbations
The Adiabatic Theorem

5. Perturbations that reach their full strength very slowly obey the so-called adiabatic theorem:
if the system is initially in an eigenstate of before the perturbation starts to change, then provided the change in occurs slowly enough, it will adiabatically follow the change in the Hamiltonian, staying in an instantaneous eigenstate of while the change is taking place, and ending in the corresponding eigenstate of the final Hamiltonian
To see this we present a "perturbative proof" of the adiabatic theorem, by dividing the interval over which the perturbation achieves full strength into a very large number N of intervals of time of duration τ = T /N, in each of which the Hamiltonian changes by at most an infinitesimal amount.

6. Perturbations that reach their full strength very slowly obey the so-called adiabatic theorem:
If the system is initially in an eigenstate of before the Hamiltonian starts to change, then provided the change in occurs slowly enough, it will adiabatically follow the change in the Hamiltonian, staying in an instantaneous eigenstate of while the change is taking place, and ending in the corresponding eigenstate of the final Hamiltonian
To see this we present a "perturbative proof" of the adiabatic theorem, by dividing the interval over which the perturbation achieves full strength into a very large number N of intervals of time of duration τ = T /N, in each of which the Hamiltonian changes by at most an infinitesimal amount.

7. Perturbations that reach their full strength very slowly obey the so-called adiabatic theorem:
If the system is initially in an eigenstate of before the Hamiltonian starts to change, then provided the change in occurs slowly enough, it will adiabatically follow the change in the Hamiltonian, staying in an instantaneous eigenstate of while the change is taking place, and ending in the corresponding eigenstate of the final Hamiltonian
To see this we present a "perturbative proof" of the adiabatic theorem, by dividing the interval over which the perturbation achieves full strength into a very large number N of intervals of time of duration τ = T /N, in each of which the Hamiltonian changes by at most an infinitesimal amount.

8. Suppose, that at the beginning of the kth such interval the system happens to be in an instantaneous eigenstate of the Hamiltonian at the beginning of that interval.
Focusing on this one time interval, we reset the scale of time so that the beginning of the interval corresponds to t = 0 and the end to t = τ .
We also will tend to suppress the index k while working within this particular time interval, writing
and denoting by the nth energy eigenvalue of

9. Suppose, that at the beginning of the kth such interval the system happens to be in an instantaneous eigenstate of the Hamiltonian at the beginning of that interval.
Focusing on this one time interval, we reset the scale of time so that the beginning of the interval corresponds to t = 0 and the end to t = τ .
We also will tend to suppress the index k while working within this particular time interval, writing
and denoting by the nth energy eigenvalue of

10. Suppose, that at the beginning of the kth such interval the system happens to be in an instantaneous eigenstate of the Hamiltonian at the beginning of that interval.
Focusing on this one time interval, we reset the scale of time so that the beginning of the interval corresponds to t = 0 and the end to t = τ .
We also will tend to suppress the index k while working within this particular time interval, writing
and denoting by the nth energy eigenvalue of

11. Suppose, that at the beginning of the kth such interval the system happens to be in an instantaneous eigenstate of the Hamiltonian at the beginning of that interval.
Focusing on this one time interval, we reset the scale of time so that the beginning of the interval corresponds to t = 0 and the end to t = τ .
We also will tend to suppress the index k while working within this particular time interval, writing
and denoting by the nth energy eigenvalue of

12. Suppose, that at the beginning of the kth such interval the system happens to be in an instantaneous eigenstate of the Hamiltonian at the beginning of that interval.
Focusing on this one time interval, we reset the scale of time so that the beginning of the interval corresponds to t = 0 and the end to t = τ .
We also will tend to suppress the index k while working within this particular time interval, writing
and denoting by the nth energy eigenvalue of

13. With this simplified notation, we then note at the end of this interval the Hamiltonian will have evolved into a new operator
where the infinitesimal change in H over this time interval is, by construction, small in the perturbative sense compared to
The slight variation in H(t) may then be expanded during this interval as
where

14. With this simplified notation, we then note at the end of this interval the Hamiltonian will have evolved into a new operator
where the infinitesimal change in H over this time interval is, by construction, small in the perturbative sense compared to
The slight variation in H(t) may then be expanded during this interval as
where

15. With this simplified notation, we then note at the end of this interval the Hamiltonian will have evolved into a new operator
where the infinitesimal change in H over this time interval is, by construction, small in the perturbative sense compared to
The slight variation in H(t) may then be expanded during this interval as
where

16. With this simplified notation, we then note at the end of this interval the Hamiltonian will have evolved into a new operator
where the infinitesimal change in H over this time interval is, by construction, small in the perturbative sense compared to
The slight variation in H(t) may then be expanded during this interval as
where

17. With this simplified notation, we then note at the end of this interval the Hamiltonian will have evolved into a new operator
where the infinitesimal change in H over this time interval is, by construction, small in the perturbative sense compared to
The slight variation in H(t) may then be expanded during this interval as
where

18. At the end of this time interval, the system has evolved to
where to lowest non-vanishing order in the perturbation
and for
in which we have introduced

19. At the end of this time interval, the system has evolved to
where to lowest non-vanishing order in the perturbation
and for
in which we have introduced

20. At the end of this time interval, the system has evolved to
where to lowest non-vanishing order in the perturbation
and for
in which we have introduced

21. At the end of this time interval, the system has evolved to
where to lowest non-vanishing order in the perturbation
and for
in which we have introduced

22. At the end of this time interval, the system has evolved to
where to lowest non-vanishing order in the perturbation
and for
in which we have introduced

23. At the end of this time interval, the system has evolved to
where to lowest non-vanishing order in the perturbation
and for
in which we have introduced

24. Evaluating the integral, we find that
Clearly, by making the change sufficiently slow (i.e., keeping the number N of intervals fixed but taking T sufficiently large) the second term can be made as small as desired.
Retaining the first term in this last expression then gives the result
where we have re-expressed in terms of the corresponding eigenvalues of H⁽⁰⁾.

25. Evaluating the integral, we find that
Clearly, by making the change sufficiently slow (i.e., keeping the number N of intervals fixed but taking T sufficiently large) the second term can be made as small as desired.
Retaining the first term in this last expression then gives the result
where we have re-expressed in terms of the corresponding eigenvalues of H⁽⁰⁾.

26. Evaluating the integral, we find that
Clearly, by making the change sufficiently slow (i.e., keeping the number N of intervals fixed but taking T sufficiently large) the second term can be made as small as desired.
Retaining the first term in this last expression then gives the result
where we have re-expressed in terms of the corresponding eigenvalues of H⁽⁰⁾.

27. Evaluating the integral, we find that
Clearly, by making the change sufficiently slow (i.e., keeping the number N of intervals fixed but taking T sufficiently large) the second term can be made as small as desired.
Retaining the first term in this last expression then gives the result
where we have re-expressed in terms of the corresponding eigenvalues of H⁽⁰⁾.

28. Thus, to this order we can write
where
is the perturbative result for the exact eigenstate of expressed as an expansion in eigenstates of
Reverting back to our orignal notation, this proves that if the system begins the kth time interval in an eigenstate of , it ends in the corresponding eigenstate of

29. Thus, to this order we can write
where
is the perturbative result for the exact eigenstate of expressed as an expansion in eigenstates of
Reverting back to our orignal notation, this proves that if the system begins the kth time interval in an eigenstate of , it ends in the corresponding eigenstate of

30. Thus, to this order we can write
where
is the perturbative result for the exact eigenstate of expressed as an expansion in eigenstates of
Reverting back to our orignal notation, this proves that if the system begins the kth time interval in an eigenstate of , it ends in the corresponding eigenstate of

31. Thus, to this order we can write
where
is the perturbative result for the exact eigenstate of expressed as an expansion in eigenstates of
Reverting back to our original notation, this proves that if the system begins the kth time interval in an eigenstate of , it ends in the corresponding eigenstate of

32. We can now repeat the process (continuously) over each (perhaps long) time interval over which the Hamiltonian changes by an infinitesimal small amount.
In this way, over many such time intervals, the system has remained in the corresponding eigenstate of the evolving Hamiltonian, which can ultimately change by a very great amount.
Provided that the change occurs sufficiently slowly, however, the state of the system will adiabatically "follow" the slowly-evolving Hamiltonian.
Thus, the probability to find the system in an eigenstate of the final Hamiltonian is zero unless , i.e., unless it started in the corresponding eigenstate of the initial Hamiltonian.

33. We can now repeat the process (continuously) over each (perhaps long) time interval over which the Hamiltonian changes by an infinitesimal small amount.
In this way, over many such time intervals, the system has remained in the corresponding eigenstate of the evolving Hamiltonian, which can ultimately change by a very great amount.
Provided that the change occurs sufficiently slowly, however, the state of the system will adiabatically "follow" the slowly-evolving Hamiltonian.
Thus, the probability to find the system in an eigenstate of the final Hamiltonian is zero unless , i.e., unless it started in the corresponding eigenstate of the initial Hamiltonian.

34. We can now repeat the process (continuously) over each (perhaps long) time interval over which the Hamiltonian changes by an infinitesimal small amount.
In this way, over many such time intervals, the system has remained in the corresponding eigenstate of the evolving Hamiltonian, which can ultimately change by a very great amount.
Provided that the change occurs sufficiently slowly, however, the state of the system will adiabatically "follow" the slowly-evolving Hamiltonian.
Thus, the probability to find the system in an eigenstate of the final Hamiltonian is zero unless , i.e., unless it started in the corresponding eigenstate of the initial Hamiltonian.

35. We can now repeat the process (continuously) over each (perhaps long) time interval over which the Hamiltonian changes by an infinitesimal small amount.
In this way, over many such time intervals, the system has remained in the corresponding eigenstate of the evolving Hamiltonian, which can ultimately change by a very great amount.
Provided that the change occurs sufficiently slowly, however, the state of the system will adiabatically "follow" the slowly-evolving Hamiltonian.
Thus, the probability to find the system in an eigenstate of the final Hamiltonian is zero unless , i.e., unless it started in the corresponding eigenstate of the initial Hamiltonian.