## 1D forced harmonic oscillator. Quick solution of non-homogeneous problem.

Posted by peeterjoot on February 19, 2010

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# Motivation.

In [1] equation (25) we have a forced harmonic oscillator equation

The solution of this equation is provided, but for fun lets derive it.

# Guts

Writing

we can rewrite the second order equation as a first order linear system

Or, with , in matrix form

The two by two matrix has the same properties as the complex imaginary, squaring to the identity matrix, so the equation to solve is now of the form

The homogeneous part of the solution is just the matrix

where is a two by one column matrix of constants. Assuming for the specific solution , and substuiting we have

This integrates directly, fixing the unknown column vector function

Thus the non-homogeneous solution takes the form

Note that . Multiplying this out, and discarding all but the second row of the matrix product gives , and Feynman’s equation (26) follows directly.

# References

[1] L.M. Brown, G.D. Carson, L.F. Locke, W.W. Spirduso, S.J. Silverman, D. Holtom, E. Fisher, J.E. Mauch, J.W. Birch, K.L. Turabian, et al. *Feynman’s thesis: A New approach to quantum theory*. Houghton Mifflin, 1954.

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