Question

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A particle of mass m is in the one-dimensional potential a. Argue, based on dimensional arguments dhat the enersy levelust be of the form With α, β, γ being numerical coefficients and c(n) being a positive numerical coefficient that depends on the energy levels. Determine the values of a, β and γ.

Answer 1

$E_{n}$ has dimension of energy as it represents n-th energy level. So dimension of $E_{n}$ is [$ML^2T^{-2}$].

As $E_{n} = c(n) \hslash^ {\alpha} m^{\beta}F^{\gamma}$ and $c(n)$ is a numerical coefficient,

So, dimension of $\hslash^ {\alpha} m^{\beta}F^{\gamma}$ is same as dimension of $E_{n}$.

We know dimension of $\hslash$ , $m$ , $F$are [$ML^2T^{-1}$] , [$M$] and [$MLT^{-2}$] respectively.

So, dimension of $\hslash^ {\alpha} m^{\beta}F^{\gamma}$ = $[ML^2T^{-1}]^{\alpha}[M]^{\beta}[MLT^{-1}]^{\gamma}$ = dimension of $E_{n}$.

$\Rightarrow$ [$M^{\alpha + \beta + \gamma}L^{2\alpha +\gamma}T^{-\alpha-2\gamma}$] = [$ML^2T^{-2}$]

Comparison of powers on both side leads to-

$\alpha +\beta +\gamma =1$

$2\alpha +\gamma = 2$

$-\alpha -2\gamma = -2$

Solving above three equations we can have,

$\alpha =2/3, \beta= -1/3, \gamma= 2/3$

Hope this helps.

Good Luck.