Determine the number of nodes corresponding to the state n = 3, for particle in a 1-D box.

Concept: 

• The particle in a box problem is one of the applications of the Schrodinger wave equation for a quantum mechanical model to a simplified system consisting of a particle moving horizontally within an infinitely deep well from which it cannot escape.
• The solutions of the Schrodinger wave equation to this problem give possible values of allowed Energy E and wave function ψ that the particle can possess.
• For solving the problems of a particle in the one-dimensional box we should have some boundary for the wave function representing the particle.
• \(ψ (0)=ψ (a)=0\) i.e. there is no probability of finding the particle outside the box.
• We solve the Schrodinger time-independent wave equation for the given \(0<x<a\) in order to find the allowed wave function and energy values for the particle inside the box.
• The allowed energy value obtained is discrete and written as:

  \({{E}_{n}}=\frac{{{n}^{2}}{{π }^{2}}{{\hbar }^{2}}}{2m{{a}^{2}}}\)where n is an integer; n = 1,2,3..

• The corresponding allowed wavefunction is given by-

 \({{ψ }_{n}}=\sqrt{\frac{2}{a}}\sin \frac{nπ x}{a}\)

• The allowed wave function ψ which when squared gives us the probability of finding the particle at a certain position within the box at a given energy level.
• The points where ψ becomes zero inside the box (except boundaries) are the nodes of the system.

Calculation:

• The state of the particle is n = 3.
• We know, the wave equation is given by:

\({{ψ }_{n}}=\sqrt{\frac{2}{a}}\sin \frac{nπ x}{a}\), for n =3, the equation will be:

\({{ψ }_{3}}=\sqrt{\frac{2}{a}}\sin \frac{3π x}{a}\)

• ψ₃ will be zero when the sin function is zero, which is 3πx/a = π or 2π 

When \({3\pi x\over a} = \pi \)

\(or, x ={a\over 3}\)

When \({3\pi x\over a} = 2\pi \) 

\(or, x ={2a\over 3}\)

• So, there are two points where the wavefunction is becoming zero, so the number of nodes is 2.