Propagator for the Harmonic Oscillator

From PhyWiki
Jump to navigation Jump to search
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
The Correspondence Principle
The Philosophy of Quantum Theory
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
Charged Particles in an Electromagnetic Field
WKB Approximation
The Heisenberg Picture: Equations of Motion for Operators
The Interaction Picture
The Virial Theorem
Commutation Relations
Angular Momentum as a Generator of Rotations in 3D
Spherical Coordinates
Eigenvalue Quantization
Orbital Angular Momentum Eigenfunctions
General Formalism
Free Particle in Spherical Coordinates
Spherical Well
Isotropic Harmonic Oscillator
Hydrogen Atom
WKB in Spherical Coordinates
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].}

Contribution From Fluctuations

We now turn our attention to the "quantum" contribution to the propagator. It is given by

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_q=\int D[y(t')]\,\exp\left [\frac{i}{\hbar}\int_{0}^{\delta t}(\tfrac{1}{2}m\dot{y}^2-\tfrac{1}{2}m\omega^2y^2)\,dt'\right ].}

Note that we changed variables in the time integral to 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-t_i;\!} this will simplify the subsequent calculations. To further simplify our notation, we introduce the quantity, 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 \delta t=t_f-t_i.\!}

Because 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'=0)=y(t'=\delta t)=0,\!} we may expand 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')\!} in a Fourier sine series:

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=1}^{\infty} a_n \sin\left(\frac{n\pi t'}{\delta t}\right)}

We now re-express the path integral in terms of the coefficients 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_n\!} of this series. One may verify, with the aid of the fact 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 \int_0^{\delta t}dt'\,\sin\left (\frac{m\pi t'}{\delta t}\right )\cos\left (\frac{n\pi t'}{\delta t}\right )=0}

for all 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 m\!} 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 n\!} 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 \int_0^{\delta t}dt'\,\sin\left (\frac{m\pi t'}{\delta t}\right )\sin\left (\frac{n\pi t'}{\delta t}\right )=\int_0^{\delta t}dt'\,\cos\left (\frac{m\pi t'}{\delta t}\right )\cos\left (\frac{n\pi t'}{\delta t}\right )=0}

if 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 m\neq n,\!} 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 K_q=C\int_{-\infty}^{\infty} da_1\,\int_{-\infty}^{\infty} da_2\,\ldots\,\exp{\left[\sum_{n=1}^{\infty}\frac{im}{2\hbar}\left(\left(\frac{n\pi}{\delta t}\right)^2-\omega^2\right)a^2_n\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 C\!} is a constant that is independent of the frequency that 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\!} . The integrals over the 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_n\!} are just Gaussians; evaluating them gives us

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_q=C'\prod_{n=1}^{\infty}\left[\left(\frac{n\pi}{\delta t}\right)^2-\omega^2\right]^{-\frac{1}{2}}= C'\prod_{n=1}^{\infty}\left(\frac{\delta t}{n\pi}\right) \prod_{n=1}^{\infty}\left[1-\left(\frac{\omega\,\delta t}{n\pi}\right)^2\right]^{-\frac{1}{2}}, }

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 C'\!} is a new constant directly related to 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\!} that is also independent of the frequency of motion. Since the first product in this expression is also independent of the frequency of motion, we will absorb it into 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'\!} , thus defining yet another constant, 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''.\!} The remaining, frequency-dependent, product then evaluates to

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_q=C''\sqrt{\frac{\omega\,\delta t}{\sin(\omega\,\delta 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 should recover the propagator for a free particle that propagates back to its initial position; using this fact, we find 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\,\delta t}}.}

We have thus determined the full "quantum" contribution to the propagator,

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_q=\sqrt{\frac{m}{2\pi i \hbar\,\delta t}}\sqrt{\frac{\omega\,\delta t}{\sin(\omega\,\delta t)}}=\sqrt{\frac{m\omega}{2\pi i \hbar \sin(\omega\,\delta t)}}, }

and therefore the full propagator,

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)=\sqrt{\frac{m\omega}{2\pi i \hbar \sin{\omega(t_f-t_i)}}}\exp\left \{\frac{im\omega}{2\hbar\sin{\omega(t_f-t_i)}}[(x_i^2+x_f^2)\cos{\omega(t_f-t_i)}-2x_ix_f]\right \}. }

Reference

Our evaluation of the "quantum" contribution to the propagator uses the method presented here: File:FeynmanHibbs H O Amplitude.pdf

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.