Quantum Two
Time Dependent Perturbations
Time Dependent Perturbations The Adiabatic Theorem
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.
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.
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.
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 .
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 .
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 .
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 .
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 .
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
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
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
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
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
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 .
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 .
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 .
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 .
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 .
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 .
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⁾.
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⁾.
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⁾.
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⁾.
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 .
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 .
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 .
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 .
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.
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.
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.
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.