Transcript powerpoint file

Quantum Two
1
2
Time Independent Approximation Methods
3
As we have seen, the task of predicting the evolution of an isolated quantum
mechanical system can be reduced to the solution of an appropriate eigenvalue
equation involving the Hamiltonian of the system.
Unfortunately, only a small number of quantum mechanical systems are
amenable to an exact solution.
Moreover, even when an exact solution to the eigenvalue problem is available, it
is often useful to understand the behavior of the system in the presence of weak
external fields that my be imposed in order to probe the structure of its
stationary states.
In these situations approximate methods are required for calculating the
eigenstates of the Hamiltonian in the presence of a perturbation that makes it
difficult or impossible to obtain an exact solution.
4
As we have seen, the task of predicting the evolution of an isolated quantum
mechanical system can be reduced to the solution of an appropriate eigenvalue
equation involving the Hamiltonian of the system.
Unfortunately, only a small number of quantum mechanical systems are
amenable to an exact solution.
Moreover, even when an exact solution to the eigenvalue problem is available, it
is often useful to understand the behavior of the system in the presence of weak
external fields that my be imposed in order to probe the structure of its
stationary states.
In these situations approximate methods are required for calculating the
eigenstates of the Hamiltonian in the presence of a perturbation that makes it
difficult or impossible to obtain an exact solution.
5
As we have seen, the task of predicting the evolution of an isolated quantum
mechanical system can be reduced to the solution of an appropriate eigenvalue
equation involving the Hamiltonian of the system.
Unfortunately, only a small number of quantum mechanical systems are
amenable to an exact solution.
Moreover, even when an exact solution to the eigenvalue problem is available, it
is often useful to understand the behavior of the system in the presence of weak
external fields that my be imposed in order to probe the structure of its
stationary states.
In these situations approximate methods are required for calculating the
eigenstates of the Hamiltonian in the presence of a perturbation that makes it
difficult or impossible to obtain an exact solution.
6
As we have seen, the task of predicting the evolution of an isolated quantum
mechanical system can be reduced to the solution of an appropriate eigenvalue
equation involving the Hamiltonian of the system.
Unfortunately, only a small number of quantum mechanical systems are
amenable to an exact solution.
Moreover, even when an exact solution to the eigenvalue problem is available, it
is often useful to understand the behavior of the system in the presence of weak
external fields that my be imposed in order to probe the structure of its
stationary states.
In these situations approximate methods are required for calculating the
eigenstates of the Hamiltonian in the presence of a perturbation that makes it
difficult or impossible to obtain an exact solution.
7
There are two general approaches commonly taken in problems of this sort.
The first, referred to as the variational method, is most commonly used to obtain
information about the ground state, and low lying excited states of the system.
The second, more systematic approach is generally referred to as timeindependent perturbation theory, of which there are two versions:
non-degenerate perturbation theory and degenerate perturbation theory.
Either of these approaches is generally applicable to any set of discrete levels and
is not necessarily restricted to the solution of the energy eigenvalue problem,
but can be applied to any observable with a discrete spectrum.
We begin with a study of the variational method, which is based upon two
variation theorems, one fairly weak, and the other of which makes a stronger and
more useful statement.
8
There are two general approaches commonly taken in problems of this sort.
The first, referred to as the variational method, is most commonly used to obtain
information about the ground state, and low lying excited states of the system.
The second, more systematic approach is generally referred to as timeindependent perturbation theory, of which there are two versions:
non-degenerate perturbation theory and degenerate perturbation theory.
Either of these approaches is generally applicable to any set of discrete levels and
is not necessarily restricted to the solution of the energy eigenvalue problem,
but can be applied to any observable with a discrete spectrum.
We begin with a study of the variational method, which is based upon two
variation theorems, one fairly weak, and the other of which makes a stronger and
more useful statement.
9
There are two general approaches commonly taken in problems of this sort.
The first, referred to as the variational method, is most commonly used to obtain
information about the ground state, and low lying excited states of the system.
The second, more systematic approach is generally referred to as timeindependent perturbation theory, of which there are two versions:
non-degenerate perturbation theory and degenerate perturbation theory.
Either of these approaches is generally applicable to any set of discrete levels and
is not necessarily restricted to the solution of the energy eigenvalue problem,
but can be applied to any observable with a discrete spectrum.
We begin with a study of the variational method, which is based upon two
variation theorems, one fairly weak, and the other of which makes a stronger and
more useful statement.
10
There are two general approaches commonly taken in problems of this sort.
The first, referred to as the variational method, is most commonly used to obtain
information about the ground state, and low lying excited states of the system.
The second, more systematic approach is generally referred to as timeindependent perturbation theory, of which there are two versions:
non-degenerate perturbation theory and degenerate perturbation theory.
Either of these approaches is generally applicable to any set of discrete levels and
is not necessarily restricted to the solution of the energy eigenvalue problem,
but can be applied to any observable with a discrete spectrum.
We begin with a study of the variational method, which is based upon two
variation theorems, one fairly weak, and the other of which makes a stronger and
more useful statement.
11
There are two general approaches commonly taken in problems of this sort.
The first, referred to as the variational method, is most commonly used to obtain
information about the ground state, and low lying excited states of the system.
The second, more systematic approach is generally referred to as timeindependent perturbation theory, of which there are two versions:
non-degenerate perturbation theory and degenerate perturbation theory.
Either of these approaches is generally applicable to any set of discrete levels and
is not necessarily restricted to the solution of the energy eigenvalue problem,
but can be applied to any observable with a discrete spectrum.
We begin with a study of the variational method, which is based upon two
variational theorems, one fairly weak, and the other of which makes a stronger
and more useful statement.
12
We note in passing that is because of this simple variational theorem that one
knows that the actual ground state energy of a many-particle system is generally
lower than that of the Hartree-Fock ground state, which approximates the actual
ground state in terms of a single direct product state.
Although this simple form of the variational theorem is already useful, it is
possible to prove an even stronger variational statement that includes the simple
bounds given above as a special case.
A statement and proof of this stronger form of the variational theorem is given in
the next segment.
13
The (Weak) Variational Theorem
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The Weak Variational Theorem - Let 𝐻 be a time-independent observable (e.g.,
the Hamiltonian) for a physical system having a discrete spectrum.
The normalized eigenstates
of 𝐻 each satisfy the eigenvalue equation
.
Assume that the eigenvalues and corresponding eigenstates have been ordered,
so that
and where we allow for the possibility that either the dimension 𝑁 of the space,
or the largest eigenvalue
is infinite.
15
The Weak Variational Theorem - Let 𝐻 be a time-independent observable (e.g.,
the Hamiltonian) for a physical system having a discrete spectrum.
The normalized eigenstates
of 𝐻 each satisfy the eigenvalue equation
.
Assume that the eigenvalues and corresponding eigenstates have been ordered,
so that
and where we allow for the possibility that either the dimension 𝑁 of the space,
or the largest eigenvalue
is infinite.
16
The Weak Variational Theorem - Let 𝐻 be a time-independent observable (e.g.,
the Hamiltonian) for a physical system having a discrete spectrum.
The normalized eigenstates
of 𝐻 each satisfy the eigenvalue equation
.
Assume that the eigenvalues and corresponding eigenstates have been ordered,
so that
and where we allow for the possibility that either the dimension 𝑁 of the space,
or the largest eigenvalue
is infinite.
17
The Weak Variational Theorem - Let 𝐻 be a time-independent observable (e.g.,
the Hamiltonian) for a physical system having a discrete spectrum.
The normalized eigenstates
of 𝐻 each satisfy the eigenvalue equation
.
Assume that the eigenvalues and corresponding eigenstates have been ordered,
so that
and where we allow for the possibility that either the dimension 𝑁 of the space,
or the largest eigenvalue
is infinite.
18
Under these rather general circumstances, if |𝜓〉 is an arbitrary normalized state
of the system it is straightforward to prove the following simple form of the
variational theorem:
The mean value of 𝐻 with respect to an arbitrary normalized state |ψ〉
is greater than or equal to the minimum eigenvalue 𝐸₀ and
is less than or equal to the maximum eigenvalue
of 𝐻. , i.e., if
then
The proof follows almost trivially upon using the expansion of 𝐻 in its own
eigenstates
19
Under these rather general circumstances, if |𝜓〉 is an arbitrary normalized state
of the system it is straightforward to prove the following simple form of the
variational theorem:
The mean value of 𝐻 with respect to an arbitrary normalized state |ψ〉
is greater than or equal to the minimum eigenvalue 𝐸₀ and
is less than or equal to the maximum eigenvalue
of 𝐻,
i.e., if
then
The proof follows almost trivially upon using the expansion of 𝐻 in its own
eigenstates
20
Under these rather general circumstances, if |𝜓〉 is an arbitrary normalized state
of the system it is straightforward to prove the following simple form of the
variational theorem:
The mean value of 𝐻 with respect to an arbitrary normalized state |ψ〉
is greater than or equal to the minimum eigenvalue 𝐸₀ and
is less than or equal to the maximum eigenvalue
of 𝐻,
i.e., if
then
then
The proof follows almost trivially upon using the expansion of 𝐻 in its own
eigenstates
21
Under these rather general circumstances, if |𝜓〉 is an arbitrary normalized state
of the system it is straightforward to prove the following simple form of the
variational theorem:
The mean value of 𝐻 with respect to an arbitrary normalized state |ψ〉
is greater than or equal to the minimum eigenvalue 𝐸₀ and
is less than or equal to the maximum eigenvalue
of 𝐻,
i.e., if
then
The proof follows almost trivially from the form of the expansion of
in its own eigenstates.
22
which we can use to express the mean value of interest in the form
We then note that each term in the last sum is itself bounded, i.e.,
so that
where we have used the assumed normalization
of the otherwise arbitrary state |𝜓〉.
23
which we can use to express the mean value of interest in the form
We then note that each term in the last sum is itself bounded, i.e.,
so that
where we have used the assumed normalization
of the otherwise arbitrary state |𝜓〉.
24
which we can use to express the mean value of interest in the form
We then note that each term in the last sum is itself bounded, i.e.,
so that
where we have used the assumed normalization
of the otherwise arbitrary state |𝜓〉.
25
which we can use to express the mean value of interest in the form
We then note that each term in the last sum is itself bounded, i.e.,
so that
where we have used the assumed normalization
of the otherwise arbitrary state |𝜓〉.
26
which we can use to express the mean value of interest in the form
We then note that each term in the last sum is itself bounded, i.e.,
so that
where we have used the assumed normalization
of the otherwise arbitrary state |𝜓〉.
27
which we can use to express the mean value of interest in the form
We then note that each term in the last sum is itself bounded, i.e.,
So, using the fact that
we find that
28
which we can use to express the mean value of interest in the form
We then note that each term in the last sum is itself bounded, i.e.,
So, using the fact that
we find that
29
which we can use to express the mean value of interest in the form
We then note that each term in the last sum is itself bounded, i.e.,
So, using the fact that
we find that
30
which we can use to express the mean value of interest in the form
We then note that each term in the last sum is itself bounded, i.e.,
So, using the fact that
we find that
31
which we can use to express the mean value of interest in the form
We then note that each term in the last sum is itself bounded, i.e.,
So, using the fact that
we find that
32
which we can use to express the mean value of interest in the form
We then note that each term in the last sum is itself bounded, i.e.,
So, using the fact that
we find that
33
which we can use to express the mean value of interest in the form
We then note that each term in the last sum is itself bounded, i.e.,
So, using the fact that
we find that
34
which we can use to express the mean value of interest in the form
We then note that each term in the last sum is itself bounded, i.e.,
So, using the fact that
Thus,
we find that
35
If 𝐻 is, in fact, the Hamiltonian of a quantum mechanical system, the variational
theorem states that the ground state energy 𝐸₀ minimizes the mean value of 𝐻
taken with respect to the normalized states of the space.
This has interesting implications.
It means, for example, that one could simply choose a random sequence of
vectors in the state space of the system and evaluate the mean value of 𝐻 with
respect to each.
The smallest value obtained then gives an upper bound for the ground state
energy of the system.
By continuing this random, or "Monte Carlo", search it is possible, in principle, to
get systematically better (i.e., lower) estimates of the ground state energy and of
the actual ground state itself.
36
If 𝐻 is, in fact, the Hamiltonian of a quantum mechanical system, the variational
theorem states that the ground state energy 𝐸₀ minimizes the mean value of 𝐻
taken with respect to the normalized states of the space.
This has interesting implications.
It means, for example, that one could simply choose a random sequence of
vectors in the state space of the system and evaluate the mean value of 𝐻 with
respect to each.
The smallest value obtained then gives an upper bound for the ground state
energy of the system.
By continuing this random, or "Monte Carlo", search it is possible, in principle, to
get systematically better (i.e., lower) estimates of the ground state energy and of
the actual ground state itself.
37
If 𝐻 is, in fact, the Hamiltonian of a quantum mechanical system, the variational
theorem states that the ground state energy 𝐸₀ minimizes the mean value of 𝐻
taken with respect to the normalized states of the space.
This has interesting implications.
It means, for example, that one could simply choose a random sequence of
vectors in the state space of the system and evaluate the mean value of 𝐻 with
respect to each.
The smallest value obtained then gives an upper bound for the ground state
energy of the system.
By continuing this random, or "Monte Carlo", search it is possible, in principle, to
get systematically better (i.e., lower) estimates of the ground state energy and of
the actual ground state itself.
38
If 𝐻 is, in fact, the Hamiltonian of a quantum mechanical system, the variational
theorem states that the ground state energy 𝐸₀ minimizes the mean value of 𝐻
taken with respect to the normalized states of the space.
This has interesting implications.
It means, for example, that one could simply choose a random sequence of
vectors in the state space of the system and evaluate the mean value of 𝐻 with
respect to each.
The smallest value obtained then gives an upper bound for the ground state
energy of the system.
By continuing this random, or "Monte Carlo", search it is possible, in principle, to
get systematically better (i.e., lower) estimates of the ground state energy and of
the actual ground state itself.
39
If 𝐻 is, in fact, the Hamiltonian of a quantum mechanical system, the variational
theorem states that the ground state energy 𝐸₀ minimizes the mean value of 𝐻
taken with respect to the normalized states of the space.
This has interesting implications.
It means, for example, that one could simply choose a random sequence of
vectors in the state space of the system and evaluate the mean value of 𝐻 with
respect to each.
The smallest value obtained then gives an upper bound for the ground state
energy of the system.
By continuing this random, or "Monte Carlo", search it is possible, in principle, to
get systematically better (i.e., lower) estimates of the ground state energy and of
the actual ground state itself.
40
We note in passing that it is because of this simple variational theorem that one
knows that the actual ground state energy of a many-particle system is generally
lower than that of the Hartree-Fock ground state, which approximates the actual
ground state in terms of a single direct product state.
Although this simple form of the variational theorem is already useful, it is
possible to prove an even stronger variational statement that includes the simple
bounds given above as a special case.
41
We note in passing that is because of this simple variational theorem that one
knows that the actual ground state energy of a many-particle system is generally
lower than that of the Hartree-Fock ground state, which approximates the actual
ground state in terms of a single direct product state.
Although this simple form of the variational theorem is already useful, it is
possible to prove an even stronger variational statement that includes the simple
bounds given above as a special case.
A statement and proof of this stronger form of the variational theorem is given in
the next segment.
42
We note in passing that is because of this simple variational theorem that one
knows that the actual ground state energy of a many-particle system is generally
lower than that of the Hartree-Fock ground state, which approximates the actual
ground state in terms of a single direct product state.
Although this simple form of the variational theorem is already useful, it is
possible to prove an even stronger variational statement that includes the simple
bounds given above as a special case.
A statement and proof of this stronger form of the variational theorem is given in
the next segment.
43
44