Electrons and Matter Waves — Question 3

Question 3

Changing particle mass gives different wavelength ratios under different constraints

Compare an electron of mass m with a neutral particle of mass $M=1840m$. Both are nonrelativistic. Use $\lambda=h/p$ and $K=p^2/(2m)$ with the appropriate mass.

1. At equal kinetic energy, find $\lambda_M/\lambda_m$ and $v_M/v_m$.

2. At equal wavelength instead, find $K_M/K_m$ and $v_M/v_m$.

Question 3 - Solution

Changing particle mass gives different wavelength ratios under different constraints

(a) Method and calculation

$$K_M=K_m:\quad p_M/p_m=\sqrt{M/m}=\sqrt{1840}.$$$$\boxed{\lambda_M/\lambda_m=v_M/v_m=1/\sqrt{1840}=.0233126}.$$

(b) Interpretation and further calculation

$$\lambda_M=\lambda_m:\quad p_M=p_m.$$$$\boxed{K_M/K_m=v_M/v_m=m/M=1/1840}.$$

Physical check

Equal energy and equal momentum are different experimental comparisons. The stated constraint must be used before applying mass scaling.