Transcript powerpoint file

Quantum One
1
2
Feynman’s Path Integral Formulation of Quantum
Mechanics
3
In the last segment we learned about different pictures of quantum mechanics.
In the Schrödinger picture, the state of the system evolves, but the basic
observables are associated with time independent Hermitian operators.
This is the picture for which our postulates in the general formulation of quantum
mechanics were developed.
But other pictures exist, including the Heisenberg picture and various Interaction
pictures, in which unitary evolution operators are used to transfer some or all of
the time dependence from the states to the observables, while preserving all
time-dependent mean values, and thereby preserving the predictions of quantum
mechanics regarding the outcomes of any measurement.
4
In this segment we develop another formulation of quantum mechanics
introduced by Feynman (while he was a graduate student!) which describes the
evolution of the system in a way which is very different from that of Schrödinger
or any of the other pictures that we have already described.
Indeed, in a certain sense, Feynman's approach constitutes a Lagrangian
approach to the evolution of quantum systems, in contrast to the Hamiltonian
approach developed by Schrödinger.
For convenience we only consider Feynman's formulation of quantum mechanics
for a single spinless particle moving in one dimension under the influence of a
time independent potential V(x) and focus, as did Feynman, on the so-called real
space propagator G(x, t | x₀, t₀) which by definition gives the quantum
mechanical amplitude to find the particle at x at time t if it was known to be at
the point x₀ at time t₀.
5
In this segment we develop another formulation of quantum mechanics
introduced by Feynman (while he was a graduate student!) which describes the
evolution of the system in a way which is very different from that of Schrödinger
or any of the other pictures that we have already described.
Indeed, in a certain sense, Feynman's approach constitutes a Lagrangian
approach to the evolution of quantum systems, in contrast to the Hamiltonian
approach developed by Schrödinger.
For convenience we only consider Feynman's formulation of quantum mechanics
for a single spinless particle moving in one dimension under the influence of a
time independent potential V(x) and focus, as did Feynman, on the so-called real
space propagator G(x, t | x₀, t₀) which by definition gives the quantum
mechanical amplitude to find the particle at x at time t if it was known to be at
the point x₀ at time t₀.
6
In this segment we develop another formulation of quantum mechanics
introduced by Feynman (while he was a graduate student!) which describes the
evolution of the system in a way which is very different from that of Schrödinger
or any of the other pictures that we have already described.
Indeed, in a certain sense, Feynman's approach constitutes a Lagrangian
approach to the evolution of quantum systems, in contrast to the Hamiltonian
approach developed by Schrödinger.
For convenience we only consider Feynman's formulation of quantum mechanics
for a single spinless particle moving in one dimension under the influence of a
time independent potential V(x) and focus, as did Feynman, on the so-called real
space propagator G(x, t | x₀, t₀) which by definition gives the quantum
mechanical amplitude to find the particle at x at time t if it was known to be at
the point x₀ at time t₀.
7
Feynman's basic postulate asserts that this propagator can be expressed as a
"path integral“
by which is meant literally (somehow) an integral over all possible paths or
trajectories that begin at x₀ at time t₀ and end at x at time t, and in which D[x(τ)]
represents the differential element or integration measure over paths, while
is the classical action evaluated for each path, defined as a time integral (along
the trajectory x(τ)) of the associated Lagrangian
8
Feynman's basic postulate asserts that this propagator can be expressed as a
"path integral“
by which is meant literally (somehow) an integral over all possible paths or
trajectories that begin at x₀ at time t₀ and end at x at time t, and in which D[x(τ)]
represents the differential element or integration measure over paths, while
is the classical action evaluated for each path, defined as a time integral (along
the trajectory x(τ)) of the associated Lagrangian
9
Feynman's basic postulate asserts that this propagator can be expressed as a
"path integral“
by which is meant literally (somehow) an integral over all possible paths or
trajectories that begin at x₀ at time t₀ and end at x at time t, and in which D[x(τ)]
represents the differential element or integration measure over paths, while
is the classical action evaluated for each path, defined as a time integral (along
the trajectory x(τ)) of the associated Lagrangian
10
Feynman's basic postulate asserts that this propagator can be expressed as a
"path integral“
by which is meant literally (somehow) an integral over all possible paths or
trajectories that begin at x₀ at time t₀ and end at x at time t, and in which D[x(τ)]
represents the differential element or integration measure over paths, while
is the classical action evaluated for each path, defined as a time integral (along
the trajectory x(τ)) of the associated Lagrangian
11
Recall that in classical mechanics one finds that the actual path taken by a particle
is one for which the action is locally stationary, i.e., for which
δS = 0.
Recall also that, through the calculus of variations this condition leads to
Lagrange's equations of motion
which for a single particle are equivalent to Newton's equations.
12
Recall that in classical mechanics one finds that the actual path taken by a particle
is one for which the action is locally stationary, i.e., for which
δS = 0.
Recall also that, through the calculus of variations this condition leads to
Lagrange's equations of motion
which for a single particle are equivalent to Newton's equations.
13
To derive Feynman's postulate from Schrödinger's mechanics we begin by
identifying the propagator as the corresponding matrix element
of the evolution operator for this conservative system,
which evolves the initial state |x₀〉 over the time interval
of interest,
and at the end of which an inner product is taken with the final state |x〉 in order
to generate the appropriate amplitude to find the particle there.
14
To derive Feynman's postulate from Schrödinger's mechanics we begin by
identifying the propagator as the corresponding matrix element
of the evolution operator for this conservative system,
which evolves the initial state |x₀〉 over the time interval
of interest,
and at the end of which an inner product is taken with the final state |x〉 in order
to generate the appropriate amplitude to find the particle there.
15
To derive Feynman's postulate from Schrödinger's mechanics we begin by
identifying the propagator as the corresponding matrix element
of the evolution operator for this conservative system,
which evolves the initial state |x₀〉 over the time interval
of interest,
and at the end of which an inner product is taken with the final state |x〉 in order
to generate the appropriate amplitude to find the particle there.
16
We note in passing that the term propagator stems from the observation that if
then
so the kernel G(x, t | x₀, t₀) propagates the wave function from whatever it was at
t₀ to whatever it is at t.
17
We note in passing that the term propagator stems from the observation that if
then
so the kernel G(x, t | x₀, t₀) propagates the wave function from whatever it was at
t₀ to whatever it is at t.
18
We note in passing that the term propagator stems from the observation that if
then
so the kernel G(x, t | x₀, t₀) propagates the wave function from whatever it was at
t₀ to whatever it is at t.
19
We note in passing that the term propagator stems from the observation that if
then
so the kernel G(x, t | x₀, t₀) propagates the wave function from whatever it was at
t₀ to whatever it is at t.
20
We note in passing that the term propagator stems from the observation that if
then
so the kernel G(x, t | x₀, t₀) propagates the wave function from whatever it was at
time t₀ to whatever it is at time t.
21
To proceed we then note that the exponential evolution operator can be factored
into many versions of itself, i.e, we can write for any integer N > 0
where
becomes as small as desired as we let N increase without bound.
22
To proceed we then note that the exponential evolution operator can be factored
into many versions of itself, i.e, we can write for any integer N > 0
where
becomes as small as desired as we let N increase without bound.
23
To proceed we then note that the exponential evolution operator can be factored
into many versions of itself, i.e, we can write for any integer N > 0
where
becomes as small as desired as we let N increase without bound.
24
To proceed we then note that the exponential evolution operator can be factored
into many versions of itself, i.e, we can write for any integer N > 0
where
becomes as small as desired as we let N increase without bound.
25
To proceed we then note that the exponential evolution operator can be factored
into many versions of itself, i.e, we can write for any integer N > 0
where
becomes as small as desired as we let N increase without bound.
26
In this limit, ignoring terms of order (Δτ)² compared to terms of first order, we
have to as good an approximation as we need
where the last line is verified by carrying out the product indicated and neglecting
any terms of second order.
27
In this limit, ignoring terms of order (Δτ)² compared to terms of first order, we
have to as good an approximation as we need
where the last line is verified by carrying out the product indicated and neglecting
any terms of second order.
28
In this limit, ignoring terms of order (Δτ)² compared to terms of first order, we
have to as good an approximation as we need
where the last line is verified by carrying out the product indicated and neglecting
any terms of second order.
29
In this limit, ignoring terms of order (Δτ)² compared to terms of first order, we
have to as good an approximation as we need
where the last line is verified by carrying out the product indicated and neglecting
any terms of second order.
30
In this limit, ignoring terms of order (Δτ)² compared to terms of first order, we
have to as good an approximation as we need
where the last line is verified by carrying out the product indicated and neglecting
any terms of second order.
31
But this implies that to this order we can write
which again is confirmed by expanding each factor on the right hand side to order
Δτ. Thus, our N-factor form for the evolution operator can also be written
where each of the
are exactly the same.
32
But this implies that to this order we can write
which again is confirmed by expanding each factor on the right hand side to order
Δτ. Thus, our N-factor form for the evolution operator can also be written
where each of the
are exactly the same.
33
But this implies that to this order we can write
which again is confirmed by expanding each factor on the right hand side to order
Δτ. Thus, our N-factor form for the evolution operator can also be written
where each of the
are exactly the same.
34
But this implies that to this order we can write
which again is confirmed by expanding each factor on the right hand side to order
Δτ. Thus, our N-factor form for the evolution operator can also be written
where each of the
are exactly the same.
35
Thus, we can write the real space propagator in the form
Inserting a complete set of position states between each of the operator factors
in the matrix element appearing in this last expression gives
where
.
36
Thus, we can write the real space propagator in the form
Inserting a complete set of position states between each of the operator factors
in the matrix element appearing in this last expression gives
where
.
37
Thus, we can write the real space propagator in the form
Inserting a complete set of position states between each of the operator factors
in the matrix element appearing in this last expression gives
where
.
38
Thus, we can write the real space propagator in the form
Inserting a complete set of position states between each of the operator factors
in the matrix element appearing in this last expression gives
where
.
39
Thus, we can write the real space propagator in the form
Inserting a complete set of position states between each of the operator factors
in the matrix element appearing in this last expression gives
where
.
40
This leads us to consider
in the last line of which we have let the operator
act to the left on the
bra
. The remaining factor can be evaluated by identifying
as the propagator for a free particle, and using the ket-bra expansion
in the momentum representation in which this operator is diagonal.
41
This leads us to consider
in the last line of which we have let the operator
act to the left on the
bra
. The remaining factor can be evaluated by identifying
as the propagator for a free particle, and using the ket-bra expansion
in the momentum representation in which this operator is diagonal.
42
This leads us to consider
in the last line of which we have let the operator
act to the left on the
bra
. The remaining factor can be evaluated by identifying
as the propagator for a free particle, and using the ket-bra expansion
in the momentum representation in which this operator is diagonal.
43
This leads us to consider
in the last line of which we have let the operator
act to the left on the
bra
. The remaining factor can be evaluated by identifying
as the propagator for a free particle, and using the ket-bra expansion
in the momentum representation in which this operator is diagonal.
44
This gives
where in the last line we have identified the plane wave factors that emerge from
the inner product of position and momentum states.
This is a Gaussian integral, with both a quadratic and linear term in k in the
exponent. We can complete the square, and perform the integral (it requires
some care since the exponents are strictly imaginary), but in the end it simply
evaluates to an "imaginary Gaussian“.
In particular, we find that . . .
45
This gives
where in the last line we have identified the plane wave factors that emerge from
the inner product of position and momentum states.
This is a Gaussian integral, with both a quadratic and linear term in k in the
exponent. We can complete the square, and perform the integral (it requires
some care since the exponents are strictly imaginary), but in the end it simply
evaluates to an "imaginary Gaussian“.
In particular, we find that . . .
46
This gives
where in the last line we have identified the plane wave factors that emerge from
the inner product of position and momentum states.
This is a Gaussian integral, with both a quadratic and linear term in k in the
exponent. We can complete the square, and perform the integral (it requires
some care since the exponents are strictly imaginary), but in the end it simply
evaluates to an "imaginary Gaussian“.
In particular, we find that . . .
47
This gives
where in the last line we have identified the plane wave factors that emerge from
the inner product of position and momentum states.
This is a Gaussian integral, with both a quadratic and linear term in k in the
exponent. We can complete the square, and perform the integral (it requires
some care since the exponents are strictly imaginary), but in the end it simply
evaluates to an "imaginary Gaussian“.
In particular, we find that . . .
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Putting this into our expression for the propagator, we find that …
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Putting this into our expression for the propagator, we find that …
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Putting this into our expression for the propagator, we find that …
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Putting this into our expression for the propagator, we find that …
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But the product of all of these exponentials can now be combined, by simply
adding up the exponents, i.e.
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But the product of all of these exponentials can now be combined, by simply
adding up the exponents, i.e.
54
But the product of all of these exponentials can now be combined, by simply
adding up the exponents, i.e.
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We now introduce further suggestive notation, by multiplying and dividing the
first term in the summand by Δτ, and defining times
.
Also, for each fixed set of values for the integration variables
we can define an effective "trajectory" by settting
, so that
In the limit that
, the time interval
becomes an infinitesimal, the
squared term on the right hand side becomes the time derivative of a function
x(τ), and the sum approaches a Riemann integral …
56
We now introduce further suggestive notation, by multiplying and dividing the
first term in the summand by Δτ, and defining times
.
Also, for each fixed set of values for the integration variables
we can define an effective "trajectory" by setting
, so that
In the limit that
, the time interval
becomes an infinitesimal, the
squared term on the right hand side becomes the time derivative of a function
x(τ), and the sum approaches a Riemann integral …
57
We now introduce further suggestive notation, by multiplying and dividing the
first term in the summand by Δτ, and defining times
.
Also, for each fixed set of values for the integration variables
we can define an effective "trajectory" by setting
, so that
In the limit that
, the time interval
becomes an infinitesimal, the
squared term on the right hand side becomes the time derivative of a function
x(τ), and the sum approaches a Riemann integral …
58
We now introduce further suggestive notation, by multiplying and dividing the
first term in the summand by Δτ, and defining times
.
Also, for each fixed set of values for the integration variables
we can define an effective "trajectory" by setting
, so that
In the limit that
, the time interval
becomes an infinitesimal, the
squared term on the right hand side becomes the time derivative of a function
x(τ), and the sum approaches a Riemann integral …
59
We now introduce further suggestive notation, by multiplying and dividing the
first term in the summand by Δτ, and defining times
.
Also, for each fixed set of values for the integration variables
we can define an effective "trajectory" by setting
, so that
In the limit that
, the time interval
becomes an infinitesimal, the
squared term on the right hand side becomes the time derivative of a function
x(τ), and the sum approaches a Riemann integral …
60
where
is the Lagrangian, and S[x(τ)] the associated action.
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where
is the Lagrangian, and S[x(τ)] the associated action.
62
where
is the Lagrangian, and S[x(τ)] the associated action.
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where
is the Lagrangian, and S[x(τ)] the associated action.
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where
is the Lagrangian, and S[x(τ)] the associated action.
65
Thus, it merely remains to identify in this limit
the corresponding integration measure, to obtain Feynman's result
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Thus, it merely remains to identify in this limit
the corresponding integration measure, to obtain Feynman's result
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It is worth noting that since the action is strictly real, the integrand simply
represents an oscillating phase factor.
In normal integrals involving oscillating phases, such as,
there is a something called the principle of stationary phase that says that, for
large λ, when the oscillations in phase are rapid, the main contributions to the
integral will come from regions where locally there is no oscillation going on, i.e.,
at the points where g(x) is locally stationary, i.e., where g′(x) = 0.
68
It is worth noting that since the action is strictly real, the integrand simply
represents an oscillating phase factor.
In normal integrals involving oscillating phases, such as,
there is a something called the principle of stationary phase that says that, for
large λ, when the oscillations in phase are rapid, the main contributions to the
integral will come from regions where locally there is no oscillation going on, i.e.,
at the points where g(x) is locally stationary, i.e., where g′(x) = 0.
69
It is worth noting that since the action is strictly real, the integrand simply
represents an oscillating phase factor.
In normal integrals involving oscillating phases, such as,
there is a something called the principle of stationary phase that says that, for
large λ, when the oscillations in phase are rapid, the main contributions to the
integral will come from regions where locally there is no oscillation going on, i.e.,
at the points where g(x) is locally stationary, the points where g′(x) = 0.
70
In the Feynman path integral, we note that the inverse power of ℏ in the
exponent is classically very small, and so in the classical limit, the path integral
has just this rapidly oscillating character.
The principle of stationary phase then suggests that the main contribution to the
integral will come from those paths for which the exponent is stationary, i.e.,
those paths for which
But these are precisely the paths predicted by classical mechanics.
Thus, remarkably, Feynman's path integral approach to quantum mechanics
implicitly allows for the derivation of all of classical mechanics as well!
71
In the Feynman path integral, we note that the inverse power of ℏ in the
exponent is classically very small, and so in the classical limit, the path integral
has just this rapidly oscillating character.
The principle of stationary phase then suggests that the main contribution to the
integral will come from those paths for which the exponent is stationary, i.e.,
those paths for which
But these are precisely the paths predicted by classical mechanics.
Thus, remarkably, Feynman's path integral approach to quantum mechanics
implicitly allows for the derivation of all of classical mechanics as well!
72
In the Feynman path integral, we note that the inverse power of ℏ in the
exponent is classically very small, and so in the classical limit, the path integral
has just this rapidly oscillating character.
The principle of stationary phase then suggests that the main contribution to the
integral will come from those paths for which the exponent is stationary, i.e.,
those paths for which
But these are precisely the paths predicted by classical mechanics.
Thus, remarkably, Feynman's path integral approach to quantum mechanics
implicitly allows for the derivation of all of classical mechanics as well!
73
In the Feynman path integral, we note that the inverse power of ℏ in the
exponent is classically very small, and so in the classical limit, the path integral
has just this rapidly oscillating character.
The principle of stationary phase then suggests that the main contribution to the
integral will come from those paths for which the exponent is stationary, i.e.,
those paths for which
But these are precisely the paths predicted by classical mechanics.
Thus, remarkably, Feynman's path integral approach to quantum mechanics
implicitly allows for the derivation of all of classical mechanics as well!
74
In the Feynman path integral, we note that the inverse power of ℏ in the
exponent is classically very small, and so in the classical limit, the path integral
has just this rapidly oscillating character.
The principle of stationary phase then suggests that the main contribution to the
integral will come from those paths for which the exponent is stationary, i.e.,
those paths for which
But these are precisely the paths predicted by classical mechanics.
Thus, remarkably, Feynman's path integral approach to quantum mechanics
implicitly allows for the derivation of all of classical mechanics as well!
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