Question

If \( y_n(x) \) is a solution of the differential equation \[ y'' - 2xy' + 2ny = 0 \] where \( n \) is an integer and the prime (\( ' \)) denotes differentiation with respect to \( x \), then acceptable plot(s) of \( \psi_n(x) = e^{-x^2/2} y_n(x) \), is(are) 

• A. A
• B. B
• C. C
• D. D

Hint:

Hermite polynomials \( H_n(x) \) form the solutions to the differential equation \( y'' - 2xy' + 2ny = 0 \), and \( \psi_n(x) \) is a Gaussian-modulated version of these polynomials.

Answer 1

The Correct Option is B, C

The given differential equation is: \[ y'' - 2xy' + 2ny = 0 \] This is a form of Hermite's differential equation, which is satisfied by the Hermite polynomials \( H_n(x) \) for integer values of \( n \). The general solution of this equation is: \[ y_n(x) = H_n(x)e^{-x^2/2} \] where \( H_n(x) \) is the n-th Hermite polynomial.
The function \( \psi_n(x) \) is given by: \[ \psi_n(x) = e^{-x^2/2} y_n(x) = H_n(x)e^{-x^2} \] The behavior of \( \psi_n(x) \) depends on the values of \( n \). 1. For \( n = 0 \): The solution is \( y_0(x) = 1 \), and thus \( \psi_0(x) = e^{-x^2/2} \). This corresponds to a Gaussian curve, which is symmetric and bell-shaped. This corresponds to option (B) in the graph, where the plot for \( n = 0 \) is a smooth Gaussian-like curve. 2. For \( n = 1 \): The solution is \( y_1(x) = -2x \), and thus \( \psi_1(x) = -2x e^{-x^2/2} \). This corresponds to a curve that starts at 0, increases, and then decreases as \( x \) increases. It also shows an oscillatory nature due to the presence of the term \( x \) multiplied by the exponential term. This corresponds to option (C) in the graph, where the plot for \( n = 1 \) is an oscillatory curve with one zero crossing. Thus, the correct answer is option (B) for \( n = 0 \) and option (C) for \( n = 1 \).