In quantum mechanics, the study of the Schrödinger

Chapter 4, Problem 49E

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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

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,

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