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10.31: Momentum-Space Variation Method for Particle in a Gravitational Field

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    The following problem deals with a particle of unit mass in a gravitational field with acceleration due to gravity equal to 1.

    Energy operator for particles near Earth's surface:

    \[ \frac{-1}{2 \mu} \frac{d^2}{dz^2} \blacksquare + z \blacksquare \nonumber \]

    Trial wave function:

    \[ \Psi ( \alpha , z) := 2 \alpha ^{ \frac{3}{2}} z exp(- \alpha z) \nonumber \]

    Fourier the position wave function into momentum space:

    \[ \Phi ( \alpha , p) := \frac{1}{ \sqrt{2 \pi}} \int_{0}^{ \infty} exp(-1 p z) \Psi ( \alpha , z) dz |_{simplify}^{assume,~ \alpha > 0} \rightarrow \frac{2 ^{ \frac{1}{2}}}{ \pi ^{ \frac{1}{2}}} \frac{ \alpha ^{ \frac{3}{2}}}{(ip + a)^2} \nonumber \]

    Demonstrate that the momentum wave function is normalized.

    \[ \int_{- \infty}^{ \infty} \overline{ \Phi ( \alpha , p)} \Phi ( \alpha , p) dp~~assume,~ \alpha > 0 \rightarrow 1 \nonumber \]

    Energy operator in momentum space:

    \[ \frac{p^2}{2} \blacksquare + i \frac{d}{dp} \blacksquare \nonumber \]

    Evaluate the variational expression for the energy:

    \[ E( \alpha ) := \int_{- \infty}^{ \infty} \overline{ \Phi ( \alpha , p)} \frac{p^2}{2} \Phi ( \alpha , p) dp ... + \int_{- \infty}^{ \infty} \overline{ \Phi ( \alpha , p)} i ( \frac{d}{dp} \Phi ( \alpha , p)) dp \nonumber \]

    Minimize energy with respect to variational parameter \( \alpha\):

    \( \alpha\) := 1 \( \alpha\) := Minimize (E, \( \alpha\) \( \alpha\) = 1.145 E( \(\alpha\)) = 1.966

    This momentum space result is in exact agreement with the coordinate-space result. The exact value for the energy is 1.856.

    \[ \frac{E ( \alpha) - 1.856}{1.856} = 5.9 \nonumber \]

    This page titled 10.31: Momentum-Space Variation Method for Particle in a Gravitational Field is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by Frank Rioux via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.