Principles of Quantum Mechanics · Problem 7.3.1

The Harmonic Oscillator — R. Shankar

Physics — Quantum Mechanics · harmonic-oscillator-power-series

Problem

^*

Consider the question why we tried a power-series solution for Eq. (7.3.11) but not Eq. (7.3.8). By feeding in a series into the latter, verify that a three-term recursion relation between

C_{n+2}

C_n

, and

C_{n-2}

obtains, from which the solution does not follow so readily. The problem is that

\psi''

has two powers of

y

less than

2\varepsilon\psi

, while the

-2yu'

piece has two more powers of

y

. In Eq. (7.3.11) on the other hand, of the three pieces

u''

-2yu'

, and

(2\varepsilon-1)u

, the last two have the same powers of

y

Intuitive explanation

The key insight is recognizing why power-series solutions work better for some differential equations than others: you need the dominant terms to have matching powers of $y$ so they can be collected and balanced together. In Eq. (7.3.8), the $\psi''$ and $-2yu'$ terms differ by two powers of $y$ , creating a three-term recursion that couples non-adjacent coefficients—much harder to solve. By contrast, Eq. (7.3.11) has $-2yu'$ and $(2\varepsilon-1)u$ at the same order, yielding a two-term recursion that closes neatly. This structural difference determines whether the series ansatz leads to a solvable recurrence.

Intuitive explanation

Related theory

Power In Ac Circuits And Power Factor

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