Quantum One 1 2 Feynmans Path Integral Formulation

Quantum One 1 2 Feynmans Path Integral Formulation of Quantum Mechanics 3

In the last segment we learned about different pictures of quantum mechanics. In the Schrdinger 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 Schrdinger 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 Schrdinger. 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 | x, t) 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 x at time t. 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 Schrdinger 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 Schrdinger. 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 | x, t) 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 x at time t. 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 Schrdinger 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 Schrdinger. 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 | x, t) 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 x at time t. 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 x at time t and end at x at time t, and in which D[x()])] 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()])) 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 x at time t and end at x at time t, and in which D[x()])] 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()])) 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 x at time t and end at x at time t, and in which D[x()])] 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()])) 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 x at time t and end at x at time t, and in which D[x()])] 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()])) 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 S = 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 S = 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 Schrdinger'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 |x 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 Schrdinger'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 |x 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 Schrdinger'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 |x 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 | x, t) propagates the wave function from whatever it was at t 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 | x, t) propagates the wave function from whatever it was at t 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 | x, t) propagates the wave function from whatever it was at t 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 | x, t) propagates the wave function from whatever it was at t 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 | x, t) propagates the wave function from whatever it was at time t 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 () 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 () 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 () 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 () 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 () 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 . 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 . 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 . 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 . 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 . . . 48 Putting this into our expression for the propagator, we find that 49

Putting this into our expression for the propagator, we find that 50 Putting this into our expression for the propagator, we find that 51 Putting this into our expression for the propagator, we find that

52 But the product of all of these exponentials can now be combined, by simply adding up the exponents, i.e. 53 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. 55 We now introduce further suggestive notation, by multiplying and dividing the first term in the summand by , 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(), 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 , 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(), 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 , 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(), 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 , 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(), 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 , 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(), and the sum approaches a Riemann integral 60 where

is the Lagrangian, and S[x()] the associated action. 61 where is the Lagrangian, and S[x()] the associated action. 62 where

is the Lagrangian, and S[x()] the associated action. 63 where is the Lagrangian, and S[x()] the associated action. 64 where

is the Lagrangian, and S[x()] the associated action. 65 Thus, it merely remains to identify in this limit the corresponding integration measure, to obtain Feynman's result 66 Thus, it merely remains to identify in this limit

the corresponding integration measure, to obtain Feynman's result 67 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 , 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 , 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 , 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 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 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 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 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 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! 75 76

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