Question

6. (Hermite polynomials) The Schrodinger equation on the real line for a simple harmonic oscillator with potential energy 1/2Kx^2, K > 0, can be written in scaled coordinates as (2.17) -y'' + x^2y = Ey, x ? ?, where E is the energy level. Here, ?? |y|^2dx < ?. a) Let y(x) = w(x)e^-x^2/2 and show that w satisfies w'' - 2xw' + (E - 1)w = 0, which is Hermite’s differential equation. b) Assume a power series solution of the form w(x) = ??0 akx^k and derive the recursion relation (k + 2)(k + 1)ak+2 = (2k + 1 - E)ak, k = 0, 1, 2, . . . . c) If E = 2n + 1, where n is a nonnegative integer, show that solutions are nth degree polynomials w(x) = Hn(x). (Hermite polynomials) Show that, appropriately normalized, the first few Hermite polynomials are H0(x) = 1, H1(x) = 2x, H2(x) = 4x^2 - 2, H3(x) = 8x^3 - 12x, . . . . d) Plot the resulting solutions y0(x),..., y3(x). (For other values of energy E the power series for w is an infinite series and behaves like ex^2; thus the normalization condition cannot hold and those energy levels are not eigenvalues.)

          6. (Hermite polynomials) The Schrodinger equation on the real line for a simple harmonic oscillator with potential energy 1/2Kx^2, K > 0, can be written in scaled coordinates as
(2.17) -y'' + x^2y = Ey, x ? ?,
where E is the energy level. Here,
?? |y|^2dx < ?.
a) Let y(x) = w(x)e^-x^2/2 and show that w satisfies
w'' - 2xw' + (E - 1)w = 0,
which is Hermite’s differential equation.
b) Assume a power series solution of the form w(x) = ??0 akx^k and derive the recursion relation
(k + 2)(k + 1)ak+2 = (2k + 1 - E)ak, k = 0, 1, 2, . . . .
c) If E = 2n + 1, where n is a nonnegative integer, show that solutions are nth degree polynomials w(x) = Hn(x). (Hermite polynomials) Show that, appropriately normalized, the first few Hermite polynomials are
H0(x) = 1, H1(x) = 2x, H2(x) = 4x^2 - 2, H3(x) = 8x^3 - 12x, . . . .
d) Plot the resulting solutions y0(x),..., y3(x). (For other values of energy E the power series for w is an infinite series and behaves like ex^2; thus the normalization condition cannot hold and those energy levels are not eigenvalues.)

6. (Hermite polynomials) The Schrodinger equation on the real line for a simple harmonic oscillator with potential energy 1/2Kx^2, K > 0, can be written in scaled coordinates as
(2.17) -y” + x^2y = Ey, x ? ?,
where E is the energy level. Here,
?? |y|^2dx < ?.
a) Let y(x) = w(x)e^-x^2/2 and show that w satisfies
w” - 2xw' + (E - 1)w = 0,
which is Hermite’s differential equation.
b) Assume a power series solution of the form w(x) = ??0 akx^k and derive the recursion relation
(k + 2)(k + 1)ak+2 = (2k + 1 - E)ak, k = 0, 1, 2, . . . .
c) If E = 2n + 1, where n is a nonnegative integer, show that solutions are nth degree polynomials w(x) = Hn(x). (Hermite polynomials) Show that, appropriately normalized, the first few Hermite polynomials are
H0(x) = 1, H1(x) = 2x, H2(x) = 4x^2 - 2, H3(x) = 8x^3 - 12x, . . . .
d) Plot the resulting solutions y0(x),..., y3(x). (For other values of energy E the power series for w is an infinite series and behaves like ex^2; thus the normalization condition cannot hold and those energy levels are not eigenvalues.)

Added by Lindsey T.

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