- #1
patric44
- 303
- 39
- Homework Statement
- i am trying to find the Asymptotic solution of the quantum harmonic oscillator using power series ?
- Relevant Equations
- y''-x^2 y = 0
hi guys
i am trying to solve the Asymptotic differential equation of the Quantum Harmonic oscillator using power series method and i am kinda stuck :
$$y'' = (x^{2}-ε)y$$
the asymptotic equation becomes :
$$y'' ≈ x^{2}y$$
using the power series method ##y(x) = \sum_{0}^{∞} a_{n}x^{n}## , this leads to :
$$\sum_{n=2}^{∞} n(n-1)a_{n}x^{n-2}+x^{2}\sum_{n=0}^{∞} (-1)a_{n}x^{n} = 0$$
$$\sum_{n=2}^{∞} n(n-1)a_{n}x^{n-2}+\sum_{n=0}^{∞} (-1)a_{n}x^{n+2} = 0$$
$$\sum_{n=2}^{∞} n(n-1)a_{n}x^{n-2}+\sum_{n=0}^{∞} (-1)a_{n}x^{n+2} = 0$$
changing n to n+4 :
$$\sum_{n=-2}^{∞} (n+4)(n+3)a_{n+4}x^{n+2}+\sum_{n=0}^{∞} (-1)a_{n}x^{n+2} = 0$$
expanding the first sum :
$$\sum_{n=0}^{∞} (n+4)(n+3)a_{n+4}x^{n+2}+\sum_{n=0}^{∞} (-1)a_{n}x^{n+2} +2a_{2}+6a_{3}x= 0$$
$$\sum_{n=0}^{∞}[(n+4)(n+3)a_{n+4}-a_{n}]x^{n+2}+2a_{2}+6a_{3}x=0$$
i am stuck here becouse i have an excess "x" term , so is this equation could be solved exactly using power series , or there is another way other than the ansatz ##e^{\pm\frac{x^{2}}{2}}## ?
i am trying to solve the Asymptotic differential equation of the Quantum Harmonic oscillator using power series method and i am kinda stuck :
$$y'' = (x^{2}-ε)y$$
the asymptotic equation becomes :
$$y'' ≈ x^{2}y$$
using the power series method ##y(x) = \sum_{0}^{∞} a_{n}x^{n}## , this leads to :
$$\sum_{n=2}^{∞} n(n-1)a_{n}x^{n-2}+x^{2}\sum_{n=0}^{∞} (-1)a_{n}x^{n} = 0$$
$$\sum_{n=2}^{∞} n(n-1)a_{n}x^{n-2}+\sum_{n=0}^{∞} (-1)a_{n}x^{n+2} = 0$$
$$\sum_{n=2}^{∞} n(n-1)a_{n}x^{n-2}+\sum_{n=0}^{∞} (-1)a_{n}x^{n+2} = 0$$
changing n to n+4 :
$$\sum_{n=-2}^{∞} (n+4)(n+3)a_{n+4}x^{n+2}+\sum_{n=0}^{∞} (-1)a_{n}x^{n+2} = 0$$
expanding the first sum :
$$\sum_{n=0}^{∞} (n+4)(n+3)a_{n+4}x^{n+2}+\sum_{n=0}^{∞} (-1)a_{n}x^{n+2} +2a_{2}+6a_{3}x= 0$$
$$\sum_{n=0}^{∞}[(n+4)(n+3)a_{n+4}-a_{n}]x^{n+2}+2a_{2}+6a_{3}x=0$$
i am stuck here becouse i have an excess "x" term , so is this equation could be solved exactly using power series , or there is another way other than the ansatz ##e^{\pm\frac{x^{2}}{2}}## ?