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In the first example, it was unclear if the leading order was in epsilon or in epsilon/delta. I clarified. |
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Next, the semiclassical approximation is used. This means that each function is expanded as a [[power series]] in {{mvar|ħ}}.
<math display="block">S = S_0 + \hbar S_1 + \hbar^2 S_2 + \cdots </math>
Substituting in the equation, and only retaining terms
<math display="block">(\nabla S_0+\hbar \nabla S_1)^2-i\hbar(\nabla^2 S_0) = 2m(E-V(\mathbf x)) </math>
which gives the following two relations:
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