3. The following boundary value problem is used in quantum mechanics to descri T electrons in conjugated linear molecules: db(x) = k²p(x); v(0) = p(L) = 0 dx? where x is position, L is the length of the molecule. a) Show that the eigenvalues are discrete. b) Find the eigenfunctions. c) Show that any two eigenfunctions, n(x) and vm(x), satisfy: Vn(x)bm(x)dx = { ( C, m= n 0, m + n where C is a constant. d) What are the dimensions of y and k?
3. The following boundary value problem is used in quantum mechanics to descri T electrons in conjugated linear molecules: db(x) = k²p(x); v(0) = p(L) = 0 dx? where x is position, L is the length of the molecule. a) Show that the eigenvalues are discrete. b) Find the eigenfunctions. c) Show that any two eigenfunctions, n(x) and vm(x), satisfy: Vn(x)bm(x)dx = { ( C, m= n 0, m + n where C is a constant. d) What are the dimensions of y and k?
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![**Problem 3**
The following boundary value problem is used in quantum mechanics to describe π electrons in conjugated linear molecules:
\[
-\frac{d^2\psi(x)}{dx^2} = k^2 \psi(x); \quad \psi(0) = \psi(L) = 0
\]
where \( x \) is position, \( L \) is the length of the molecule.
a) Show that the eigenvalues are discrete.
b) Find the eigenfunctions.
c) Show that any two eigenfunctions, \( \psi_n(x) \) and \( \psi_m(x) \), satisfy:
\[
\int_0^L \psi_n(x) \psi_m(x) dx =
\begin{cases}
C^2, & m = n \\
0, & m \neq n
\end{cases}
\]
where \( C \) is a constant.
d) What are the dimensions of \( \psi \) and \( k \)?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4f6272ad-1a25-4a67-b389-4d6bf92e8dd2%2Fda4743cf-4900-4e8b-b39b-169682dacbdf%2Fu9ksyvh_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem 3**
The following boundary value problem is used in quantum mechanics to describe π electrons in conjugated linear molecules:
\[
-\frac{d^2\psi(x)}{dx^2} = k^2 \psi(x); \quad \psi(0) = \psi(L) = 0
\]
where \( x \) is position, \( L \) is the length of the molecule.
a) Show that the eigenvalues are discrete.
b) Find the eigenfunctions.
c) Show that any two eigenfunctions, \( \psi_n(x) \) and \( \psi_m(x) \), satisfy:
\[
\int_0^L \psi_n(x) \psi_m(x) dx =
\begin{cases}
C^2, & m = n \\
0, & m \neq n
\end{cases}
\]
where \( C \) is a constant.
d) What are the dimensions of \( \psi \) and \( k \)?
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