Question

For a particle in a 1D infinite square well of width \(L\), the energy levels are proportional to
• \(n\)
• \(n^2\)
• \(\dfrac{1}{n}\)
• \(\dfrac{1}{n^2}\)

• A. \(n\)
• B. \(n^2\)
• C. \(\dfrac{1}{n}\)
• D. \(\dfrac{1}{n^2}\)

Hint:

For a particle in a 1D box: \[ E_n=\frac{n^2h^2}{8mL^2} \] Hence: \[ E_n\propto n^2 \]

Answer 1

The Correct Option is B

Concept: A particle confined in a one-dimensional infinite potential well can possess only discrete energy values. This is a direct consequence of: \[ \boxed{ \text{Quantum confinement} } \] The allowed energies are quantized.

Step 1: Write the energy expression for a particle in a box. The energy levels are given by: :contentReference[oaicite:2]{index=2} where:
• \(E_n\) = energy of \(n^{th}\) level
• \(n = 1,2,3,\dots\)
• \(h\) = Planck's constant
• \(m\) = mass of particle
• \(L\) = width of box

Step 2: Determine proportionality. From the equation: \[ E_n=\frac{n^2h^2}{8mL^2} \] All quantities except \(n\) are constants. Therefore: \[ \boxed{ E_n \propto n^2 } \] Thus energy levels increase as the square of quantum number.

Step 3: Understand physically. The first few energy levels are: \[ E_1 \propto 1^2=1 \] \[ E_2 \propto 2^2=4 \] \[ E_3 \propto 3^2=9 \] Thus: \[ E_1:E_2:E_3=1:4:9 \] Hence energy spacing increases rapidly.

Step 4: Analyze options carefully.
• \(E_n \propto n\) \(\rightarrow\) incorrect
• \(E_n \propto n^2\) \(\rightarrow\) correct
• \(E_n \propto \frac1n\) \(\rightarrow\) incorrect
• \(E_n \propto \frac1{n^2}\) \(\rightarrow\) incorrect

Step 5: Choose the correct answer. Therefore: \[ \boxed{ E_n \propto n^2 } \] Hence the correct option is: \[ \boxed{(2)} \] Additional Understanding: The particle cannot possess zero energy because: \[ n=0 \] is not allowed. Thus even the ground state possesses finite energy called: \[ \boxed{ \text{Zero point energy} } \] Final Conclusion: Energy levels in a 1D infinite square well are proportional to: \[ \boxed{ n^2 } \] Hence, the correct answer is: \[ \boxed{(2)} \]