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Anonymous asked in Science & MathematicsPhysics · 9 years ago

What is the corresponding potential function V(x)?

If a solution to Schrodinger’s equation is the wavefunction ψ(x) = A e−kx^2 (-∞ < x < ∞; k > 0) for some energy E. Find the corresponding potential function V(x)? Show Work.

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  • 9 years ago
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    Answer … V(x) = E - ( ħ ² k / m ) [ 1 - 2 k x ² ] ...

    Since … - ( ħ ² / 2m ) d ² ψ / d x ² + V ψ = E ψ …

    … V ψ = E ψ + ( ħ ² / 2m ) d ² ψ / d x ² …

    … ψ = A exp ( - k x ² ) … --> … d ψ / d x = A ( -2 k x ) exp ( - k x ² ) …

    … d ² ψ / d x ² = A [ -2 k exp ( - k x ² ) + ( -2 k x ) ² exp ( - k x ² ) ]

    ………………. = A [ -2 k + 4 k ² x ² ] exp ( - k x ² )

    ………………. = - 2 k [ 1 - 2 k x ² ] A exp ( - k x ² )

    ………………. = - 2 k [ 1 - 2 k x ² ] ψ

    … V ψ = E ψ + ( ħ ² / 2m ) d ² ψ / d x ² = E ψ + ( ħ ² / 2m ) { - 2 k [ 1 - 2 k x ² ] ψ }

    … V = E + ( ħ ² / 2m ) { - 2 k [ 1 - 2 k x ² ] }

    …… = E - ( ħ ² k / m ) [ 1 - 2 k x ² ]

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