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In quantum mechanics, the study of the Schrödinger
Chapter 4, Problem 49E(choose chapter or problem)
In quantum mechanics, the study of the Schrödinger equation for the case of a harmonic oscillator leads to a consideration of Hermite's equation,
\(y^{\prime \prime}-2 t y^{\prime}+\lambda y=0\) ,
where \(\lambda\) is a parameter. Use the reduction of order formula to obtain an integral representation of a second linearly independent solution to Hermite's equation for the given value of \(\lambda\) and corresponding solution \(f(t)\).
(a) \(\lambda=4\) , \(f(t)=1-2 t^{2}\)
(b) \(\lambda=6\) , \(f(t)=3-2 t^{3}\)
Equation Transcription:
Text Transcription:
y''-2ty'+lambda y=0
lambda
lambda
f(t)
lambda=4
f(t)=1-2t^2
lambda=6
f(t)=3-2t^3
Questions & Answers
QUESTION:
In quantum mechanics, the study of the Schrödinger equation for the case of a harmonic oscillator leads to a consideration of Hermite's equation,
\(y^{\prime \prime}-2 t y^{\prime}+\lambda y=0\) ,
where \(\lambda\) is a parameter. Use the reduction of order formula to obtain an integral representation of a second linearly independent solution to Hermite's equation for the given value of \(\lambda\) and corresponding solution \(f(t)\).
(a) \(\lambda=4\) , \(f(t)=1-2 t^{2}\)
(b) \(\lambda=6\) , \(f(t)=3-2 t^{3}\)
Equation Transcription:
Text Transcription:
y''-2ty'+lambda y=0
lambda
lambda
f(t)
lambda=4
f(t)=1-2t^2
lambda=6
f(t)=3-2t^3
ANSWER:
Solution
Step 1
Here, it is given that
So,