Propagator for the Harmonic Oscillator: Difference between revisions

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==Contribution from Classical Path==
==Contribution from Classical Path==


The classical action <math>S</math> can be evaluated as follows:
We will begin by evaluating the "classical" contribution to the propagator.  This is essentially just a problem of classical mechanics; we begin by solving for the classical motion of the particle.  The equation of motion is, as stated earlier,


<math>S=\int_{0}^{t}(KE-PE)dt </math>
<math>\ddot{x}_c+\omega^2x_c=0.\!</math>


Where <math>KE\!</math> is the kinetic engergy and <math>PE\!</math> is the potential energy.
We impose the boundary conditions, <math>x(t_i)=x_i\!</math> and <math>x(t_f)=x_f.\!</math> The solution of the equation of motion that satisfies these boundary conditions is
Equation of motion for harmonic oscillator:
<math>x_{cl}(t')=A\cos(\omega t')+B\sin(\omega t')\!</math><br/>
<math>A\!</math> and <math>B\!</math> are constants.
At <math>t'=0\!</math> (starting point),<math>x_{cl}(0)=x_0\rightarrow A=x_0</math>.


At  <math>t'=t\!</math> (final point), <math>x_{cl}(t)=x\rightarrow B=\frac{x-x_0\cos(\omega t)}{\sin(\omega t)}</math>                                                                                . </math>  
<math>x_c(t)=x_i\frac{\sin{\omega(t_f-t)}}{\sin{\omega(t_f-t_i)}}+x_f\frac{\sin{\omega(t-t_i)}}{\sin{\omega(t_f-t_i)}},</math>


is a symbolic way of saying "integrate over all paths connecting <math>{x_{0}}</math> and <math>{x_{N}}</math> (in the interval <math>{t_{0}}</math> and <math>{t_{N}}</math>)." Now, a path <math>{x_{t}}</math> is fully specified by an infinity of numbers <math>{x(t_{0})}</math>,..., <math>{x(t)}</math>, ...,<math>{x(t_{N})}</math>, namely, the values of the function <math>{x(t)}</math> at every point <math>{t}</math> is the interval <math>{t_{0}}</math> to <math>{t_{N}}</math>. To sum over all paths, we must integrate over all possible values of these infinite variables, except of course <math>{x(t_{0})}</math> and <math>{x(t_{N})}</math>, which will be kept fixed at <math>{x_{0}}</math> and <math>{x_{N}}</math>, respectively. To tackle this problem, we follow this idea that was used in section 1.10: we trade the function <math>{x_{t}}</math> for a discrete approximation which agrees with <math>{x_{t}}</math> at the <math>{N+1}</math> points.
and the corresponding velocity is
Substitute:


<math>x_{cl}(t')= x_0\cos(\omega t')+\frac{x-x_0\cos(\omega t)}{\sin(\omega t)}\sin(\omega t')
<math>x_c(t)=-\omega x_i\frac{\cos{\omega(t_f-t)}}{\sin{\omega(t_f-t_i)}}+\omega x_f\frac{\cos{\omega(t-t_i)}}{\sin{\omega(t_f-t_i)}}.</math>
\Rightarrow \frac{dx_{cl}(t')}{dt'}= -\omega x_0\sin(\omega t')+\omega \frac{x-x_0\cos(\omega t)}{\sin(\omega t)}\cos(\omega t')</math>                                                        <br/>
<math>KE= \frac{1}{2}m\left(\frac{dx_{cl}}{dt}\right)^2=\frac{1}{2}m\left[-\omega x_0\sin(\omega t')+\omega \frac{x-x_0\cos(\omega t)}{\sin(\omega t)}\cos(\omega t')\right]^2</math><br/>
<math>PE= \frac{1}{2}k(x_{cl}(t'))^2=\frac{1}{2}k\left[x_0\cos(\omega t')+\frac{x-x_0\cos(\omega t)}{\sin(\omega t)}\sin(\omega t')\right]^2</math>                                        <br/> 
Substituting, integrating from time 0 to time <math> t \! </math> and simplifying, we get:


<math>S=S(t,x,x_0)=\frac{m\omega}{2\sin(\omega t)}((x^2+x_0^2)\cos(\omega t)-2xx_0)</math><br/>
If we now substitute these expressions into the Lagrangian and simplify, we obtain
 
<math>L=\frac{m\omega^2}{2\sin^2{\omega(t_f-t_i)}}\{(x_i^2+x_f^2)\cos[2\omega(t-t_i)]-2x_ix_f\cos{\omega(t_i+t_f-2t)}\}.</math>
 
If we now substitute this into the action, we finally obtain
 
<math>S_c=\frac{m\omega}{2\sin{\omega(t_f-t_i)}}((x_i^2+x_f^2)\cos{\omega(t_f-t_i)}-2x_ix_f).</math>


Explicit evaluation of the path integral for the harmonic oscillator can be found here [[Image:FeynmanHibbs_H_O_Amplitude.pdf]]
Explicit evaluation of the path integral for the harmonic oscillator can be found here [[Image:FeynmanHibbs_H_O_Amplitude.pdf]]

Revision as of 12:47, 18 January 2014

Quantum Mechanics A
SchrodEq.png
Schrödinger Equation
The most fundamental equation of quantum mechanics; given a Hamiltonian Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal{H}} , it describes how a state Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |\Psi\rangle} evolves in time.
Basic Concepts and Theory of Motion
UV Catastrophe (Black-Body Radiation)
Photoelectric Effect
Stability of Matter
Double Slit Experiment
Stern-Gerlach Experiment
The Principle of Complementarity
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Brief Derivation of Schrödinger Equation
Relation Between the Wave Function and Probability Density
Stationary States
Heisenberg Uncertainty Principle
Some Consequences of the Uncertainty Principle
Linear Vector Spaces and Operators
Commutation Relations and Simultaneous Eigenvalues
The Schrödinger Equation in Dirac Notation
Transformations of Operators and Symmetry
Time Evolution of Expectation Values and Ehrenfest's Theorem
One-Dimensional Bound States
Oscillation Theorem
The Dirac Delta Function Potential
Scattering States, Transmission and Reflection
Motion in a Periodic Potential
Summary of One-Dimensional Systems
Harmonic Oscillator Spectrum and Eigenstates
Analytical Method for Solving the Simple Harmonic Oscillator
Coherent States
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Feynman Path Integrals
The Free-Particle Propagator
Propagator for the Harmonic Oscillator
Differential Cross Section and the Green's Function Formulation of Scattering
Central Potential Scattering and Phase Shifts
Coulomb Potential Scattering

We will now evaluate the propagator for the harmonic oscillator. The Lagrangian for this system is

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L=\tfrac{1}{2}m\dot{x}^2-\tfrac{1}{2}m\omega^2x^2.}

Before we begin, let us prove that the propagator will separate into two factors; one of these comes entirely from the classical motion of the system, and the other comes entirely from quantum fluctuations about said trajectory. To this end, let us write Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x=x_c+y,\!} where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_c\!} is the classical trajectory and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y\!} is the fluctuation, which will be a new integration variable for the path integral. If we take Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t_i\!} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t_f\!} to be the initial and final times, respectively, then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y(t_i)=y(t_f)=0.\!} Substituting this into the action, we get

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S=\int_{t_i}^{t_f} dt\,[\tfrac{1}{2}m(\dot{x}_c+\dot{y})^2-\tfrac{1}{2}m\omega^2(x_c+y)^2].}

We now expand out the squares, obtaining

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S=\int_{t_i}^{t_f} dt\,(\tfrac{1}{2}m\dot{x}_c^2-\tfrac{1}{2}m\omega^2x_c^2)+\int_{t_i}^{t_f} dt\,(\tfrac{1}{2}m\dot{y}^2-\tfrac{1}{2}m\omega^2y^2)+\int_{t_i}^{t_f} dt\,(m\dot{x}_c\dot{y}-m\omega^2x_cy).}

If we integrate by parts in the third term, we get

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S=\int_{t_i}^{t_f} dt\,(\tfrac{1}{2}m\dot{x}_c^2-\tfrac{1}{2}m\omega^2x_c^2)+\int_{t_i}^{t_f} dt\,(\tfrac{1}{2}m\dot{y}^2-\tfrac{1}{2}m\omega^2y^2)-\int_{t_i}^{t_f} dt\,m(\ddot{x}_c+\omega^2x_c)y.}

We know, however, that the classical motion obeys the equation, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ddot{x}_c+\omega^2x_c=0.\!} As a result, the third term is zero, and the action separates into two contributions, one coming entirely from the classical motion, and the other coming entirely from quantum fluctuations. Denoting these two contributions as

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S_c(x_c)=\int_{t_i}^{t_f} dt\,(\tfrac{1}{2}m\dot{x}_c^2-\tfrac{1}{2}m\omega^2x_c^2)}

and

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S_q(y)=\int_{t_i}^{t_f} dt\,(\tfrac{1}{2}m\dot{y}^2-\tfrac{1}{2}m\omega^2y^2),\!}

the propagator may now be written as

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle K(x_f,t_f;x_i,t_i)=e^{iS_c/\hbar}\int D[y(t)]\,e^{iS_q/\hbar}.}

We will now evaluate each of these contributions.

Contribution from Classical Path

We will begin by evaluating the "classical" contribution to the propagator. This is essentially just a problem of classical mechanics; we begin by solving for the classical motion of the particle. The equation of motion is, as stated earlier,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ddot{x}_c+\omega^2x_c=0.\!}

We impose the boundary conditions, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x(t_i)=x_i\!} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x(t_f)=x_f.\!} The solution of the equation of motion that satisfies these boundary conditions is

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_c(t)=x_i\frac{\sin{\omega(t_f-t)}}{\sin{\omega(t_f-t_i)}}+x_f\frac{\sin{\omega(t-t_i)}}{\sin{\omega(t_f-t_i)}},}

and the corresponding velocity is

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_c(t)=-\omega x_i\frac{\cos{\omega(t_f-t)}}{\sin{\omega(t_f-t_i)}}+\omega x_f\frac{\cos{\omega(t-t_i)}}{\sin{\omega(t_f-t_i)}}.}

If we now substitute these expressions into the Lagrangian and simplify, we obtain

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L=\frac{m\omega^2}{2\sin^2{\omega(t_f-t_i)}}\{(x_i^2+x_f^2)\cos[2\omega(t-t_i)]-2x_ix_f\cos{\omega(t_i+t_f-2t)}\}.}

If we now substitute this into the action, we finally obtain

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S_c=\frac{m\omega}{2\sin{\omega(t_f-t_i)}}((x_i^2+x_f^2)\cos{\omega(t_f-t_i)}-2x_ix_f).}

Explicit evaluation of the path integral for the harmonic oscillator can be found here File:FeynmanHibbs H O Amplitude.pdf

Contribution From Fluctuations

Now, let's evaluate the path integral: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A=A(t)=\int_{y(0)=0}^{y(t)=0}D[y(t')]e^{\frac{i}{\hbar}\int_{0}^{t}(\frac{1}{2}my'^2-\frac{1}{2}ky^2)dt'}}

Note that the integrand is taken over all possible trajectory starting at point Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_0} at time Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t'=0} , ending at point Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} at time Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t'=t} .

Expanding this integral,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A(t)=\left(\frac{m}{2\pi i \hbar}\right)^{\frac{N}{2}}\int_{-\infty}^{\infty} dy_1\ldots dy_{N-1} e^{\left[\frac{i}{\hbar}\left(\frac{m}{2\Delta t}y^2_{N-1}- \frac{\Delta t}{2}ky^2_{N-1}\right)\right]}e^{\left[\frac{i}{\hbar}\left(\frac{m}{2\Delta t}(y_{N-1}-y_{N-2})^2- \frac{\Delta t}{2}ky^2_{N-2}\right)\right]}\ldots e^{\left[\frac{i}{\hbar}\left(\frac{m}{2\Delta t}y^2_{1}- \frac{\Delta t}{2}ky^2_{1}\right)\right]} }

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N\Delta t=t\!} .

Expanding the path trajectory in Fourier series, we have Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y(t')=\sum_n a_n \sin\left(\frac{n\pi t'}{t}\right) }

we may express Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A(t)\!} in the form

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A(t)=C\int_{-\infty}^{\infty} da_1\ldots da_{N-1} \exp{\left[\sum_{n=1}^{N-1}\frac{im}{2\hbar}\left(\left(\frac{n\pi}{t}\right)^2- \omega^2\right)a^2_n\right]} }

where C is a constant independent of the frequency which comes from the Jacobian of the transformation. The important point is that it does not depend on the frequency Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega\!} . Thus, evaluating the integral of,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A(t)=C'\prod_{n=1}^{N-1}\left[\left(\frac{n\pi}{t}\right)^2-\omega^2\right]^{-\frac{1}{2}}= C'\prod_{n=1}^{N-1}\left[\left(\frac{n\pi}{t}\right)^2\right]^{-\frac{1}{2}} \prod_{n=1}^{N-1}\left[1-\left(\frac{\omega t}{n\pi}\right)^2\right]^{-\frac{1}{2}} }

where C' is a constant directly related to C and still independent of the frequency of motion. Since the first product series in this final expression is also independent of the frequency of motion, we can absorb it into our constant C' to have a new constant, C. Simplifying further,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A(t)=C''\sqrt{\frac{\omega t}{\sin(\omega t)}} }

In the limit Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega\rightarrow 0} , we already know that

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C''=\sqrt{\frac{m}{2\pi i \hbar t}} }

Thus,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A(t)=\sqrt{\frac{m}{2\pi i \hbar t}}\sqrt{\frac{\omega t}{\sin(\omega t)}}= \sqrt{\frac{m}{2\pi i \hbar \sin(\omega t)}} }

and

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle <x|\hat{U}(t,0)|x_0>=\sqrt{\frac{m}{2\pi i \hbar \sin(\omega t)}} e^{\frac{i}{\hbar}\left(\frac{m\omega}{2sin(\omega t)}((x^2+x_0^2)cos(\omega t)-2xx_0)\right)} }

Reference

For a more detailed evaluation of this problem, please see Barone, F. A.; Boschi-Filho, H.; Farina, C. 2002. "Three methods for calculating the Feynman propagator". American Association of Physics Teachers, 2003. Am. J. Phys. 71 (5), May 2003. pp 483-491.

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