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25 lines
847 B
Plaintext
25 lines
847 B
Plaintext
Show that ͟d͟〈͟p͟〉͟ = -〳͟d͟V͟(͟x͟)͟〵 when a particle is subjected to a potential 〈V(x)〉.
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dt 〵dx 〳
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The time derivative of the expectation value of the momentum is a known quantity, from
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Time Dependence of Expectation Value of General Momentum Operator:
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d〈p〉 = 1.
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dt ιħ
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The problem is therefore reduced to finding whether -/dV(x)\ reduces to 1.
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\dx / ιħ
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-/dV(x)\ = -〈Ψ| d V(x) |Ψ〉.
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\dx / dx
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Viewing the expression in this form reveals a relationship between the space derivative and the operators V(x) and |Ψ〉. The chain rule allows this derivative to be computed.
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-〈Ψ| d V(x) |Ψ〉 = -〈Ψ| ⎛d V(x)|Ψ> + d |Ψ> V(x)⎞.
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dx ⎝dx dx ⎠
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