More Electron Traps — Question 5

Question 5

Effective mass changes both level spacing and wave-function size in a fixed trap

Two materials have the same parabolic confining potential $V=Kx^2/2$, but electron effective masses $m_A=m$ and $m_B=m/16$. Treat each with $\omega=\sqrt{K/m^*}$ and $\sigma_x=\sqrt{\hbar/(2m^*\omega)}$.

1. Find the ratio of adjacent-level spacings.

2. Find the ratio of ground-state position widths and explain their different mass powers.

Question 5 - Solution

Effective mass changes both level spacing and wave-function size in a fixed trap

(a) Method and calculation

$$\Delta E=\hbar\sqrt{K/m^*}\Rightarrow\boxed{\Delta E_B/\Delta E_A=\sqrt{16}=4}.$$$$\sigma_x^2=\frac{\hbar}{2\sqrt{m^*K}}.$$

(b) Interpretation and further calculation

$$\sigma_x\propto(m^*)^{-1/4}\Rightarrow\boxed{\sigma_{x,B}/\sigma_{x,A}=16^{1/4}=2}.$$$$\boxed{E_{0,B}/E_{0,A}=4},\qquad\boxed{\sigma_{x,B}^2/\sigma_{x,A}^2=4}.$$

Physical check

These ratios hold at fixed potential curvature. Keeping the oscillator frequency fixed instead would require changing the curvature and give a different comparison.