Quantum Chemistry/Example 3

Write a question about calculating the number of nodes in a particle in a 1D box wavefunction.

Question 3

Calculate the number of nodes for a particle in a 1D box when n=2 and n=3 when the length of the box is L=5, and give the x-intercept of the node(s).

 Wavefunction for a particle in a 1D-Box ${\displaystyle \psi (x)={\sqrt {\frac {2}{L}}}\sin \left({\frac {n\pi x}{L}}\right)}$

Part 1

nnodes=n-1

nnodes=2-1=1

We're looking for one x-intercept.

${\displaystyle 0\div {\sqrt {\frac {2}{5}}}={\sqrt {\frac {2}{5}}}\sin \left({\frac {2\pi x}{5}}\right)\div {\sqrt {\frac {2}{5}}}}$

${\displaystyle 0={\sqrt {\frac {2}{5}}}\sin \left({\frac {2\pi x}{5}}\right)}$

Use the form ${\displaystyle \sin(kx)}$ where ${\displaystyle k=\left({\frac {2\pi }{5}}\right)}$

${\displaystyle Period=\left({\frac {2\pi }{k}}\right)}$

${\displaystyle Period=\left({\frac {2\pi }{\frac {2\pi }{5}}}\right)=5}$

${\displaystyle \left({\frac {Period}{2}}\right)}$ is the x-intercept

x-intercepts are 0, ${\displaystyle \left({\frac {5}{2}}\right)}$, 5

0 and 5 are the edges of the box and ${\displaystyle \left({\frac {5}{2}}\right)}$ is the only node.

Part 2

nnodes=n-1

nnodes=3-1=2

We're looking for two x-intercepts.

${\displaystyle 0\div {\sqrt {\frac {3}{5}}}={\sqrt {\frac {3}{5}}}\sin \left({\frac {3\pi x}{5}}\right)\div {\sqrt {\frac {3}{5}}}}$

${\displaystyle 0={\sqrt {\frac {3}{5}}}\sin \left({\frac {3\pi x}{5}}\right)}$

${\displaystyle k=\left({\frac {3\pi }{5}}\right)}$ ${\displaystyle Period=\left({\frac {3\pi }{\frac {3\pi }{5}}}\right)={\frac {10}{3}}}$

x-intercepts are ${\displaystyle \left({\frac {5n}{3}}\right)}$

x-intercepts = 0, ${\displaystyle \left({\frac {5}{3}}\right),\left({\frac {10}{3}}\right),5}$

0 and 5 are the edges of the box, ${\displaystyle \left({\frac {5}{3}}\right)}$ and ${\displaystyle \left({\frac {10}{3}}\right)}$ are the nodes.