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paul159753

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- I'm looking for help for an exercise of the chapter III. Schrödinger's equation, §25 The transmission coefficient (problem 3).

Hello everyone,

I'm looking for help for the problem 3 of the chapter III. Schrödinger's equation, §25 The transmission coefficient of the Volume 3 of the Landau-Lifshitz book (non-relativistic QM).

In this exercise Landau considers a smooth potential wall $$\frac{U_0}{1 + \exp{\left(-\alpha x \right)}}.$$ We search solutions for ##E > U_0##.

So we have to solve $$\frac{d^2}{dx^2}\Psi + \frac{2m}{\hbar^2}\left(E - \frac{U_0}{1 + \exp{\left(-\alpha x \right)}}\right)\Psi = 0.$$ We do the change of variable ##\xi = -\exp{\left(-\alpha x \right)}##. We write the derivative operator $$\frac{d^2}{dx^2}= \left(\frac{d\xi}{dx}\frac{d}{d\xi}\right)^2 = \alpha^2 \xi \frac{d}{d\xi} + \alpha^2 \xi ^2 \frac{d^2}{d\xi^2}.$$ Noting ##k_2^2 = 2m(E-U_0)/\hbar^2,\ k_1^2 = 2mE/\hbar^2## we can rewrite the equation $$\frac{d^2}{dx^2}\Psi + \left(\frac{k_2^2 - \xi k_1 ^2}{1 - \xi}\right)\Psi = 0.$$ We look for solutions of the form ##\Psi = \xi ^{-ik_2/\alpha} \omega(\xi)##. Multiplying the equation by ##1-\xi## and dividing by ##\alpha^2 \xi \xi ^{-ik_2/\alpha}## we get the following equation for ##\omega(\xi)## : $$ \xi(1-\xi)\frac{d^2}{d\xi^2}\omega(\xi) + (1-\xi)\left(1 - 2ik_2 / \alpha\right)\frac{d}{d\xi}\omega(\xi) + \left((k_2^2 - k_1^2)/\alpha^2 - ik_2/\alpha \frac{(1-\xi)}{\xi} \right)\omega(\xi) = 0.$$

My problem is that Landau found $$\xi(1-\xi)\frac{d^2}{d\xi^2}\omega(\xi) + (1-\xi)\left(1 - 2ik_2 / \alpha\right)\frac{d}{d\xi}\omega(\xi) + \left((k_2^2 - k_1^2)/\alpha^2 \right)\omega(\xi) = 0, $$ so the same thing but without the ##- ik_2/\alpha \frac{(1-\xi)}{\xi}\omega(\xi)## term. So he gets an hypergeometric equation and I don't. I don't know where my mistakes are.

Any help would be much appreciated.

Thanks !

I'm looking for help for the problem 3 of the chapter III. Schrödinger's equation, §25 The transmission coefficient of the Volume 3 of the Landau-Lifshitz book (non-relativistic QM).

In this exercise Landau considers a smooth potential wall $$\frac{U_0}{1 + \exp{\left(-\alpha x \right)}}.$$ We search solutions for ##E > U_0##.

So we have to solve $$\frac{d^2}{dx^2}\Psi + \frac{2m}{\hbar^2}\left(E - \frac{U_0}{1 + \exp{\left(-\alpha x \right)}}\right)\Psi = 0.$$ We do the change of variable ##\xi = -\exp{\left(-\alpha x \right)}##. We write the derivative operator $$\frac{d^2}{dx^2}= \left(\frac{d\xi}{dx}\frac{d}{d\xi}\right)^2 = \alpha^2 \xi \frac{d}{d\xi} + \alpha^2 \xi ^2 \frac{d^2}{d\xi^2}.$$ Noting ##k_2^2 = 2m(E-U_0)/\hbar^2,\ k_1^2 = 2mE/\hbar^2## we can rewrite the equation $$\frac{d^2}{dx^2}\Psi + \left(\frac{k_2^2 - \xi k_1 ^2}{1 - \xi}\right)\Psi = 0.$$ We look for solutions of the form ##\Psi = \xi ^{-ik_2/\alpha} \omega(\xi)##. Multiplying the equation by ##1-\xi## and dividing by ##\alpha^2 \xi \xi ^{-ik_2/\alpha}## we get the following equation for ##\omega(\xi)## : $$ \xi(1-\xi)\frac{d^2}{d\xi^2}\omega(\xi) + (1-\xi)\left(1 - 2ik_2 / \alpha\right)\frac{d}{d\xi}\omega(\xi) + \left((k_2^2 - k_1^2)/\alpha^2 - ik_2/\alpha \frac{(1-\xi)}{\xi} \right)\omega(\xi) = 0.$$

My problem is that Landau found $$\xi(1-\xi)\frac{d^2}{d\xi^2}\omega(\xi) + (1-\xi)\left(1 - 2ik_2 / \alpha\right)\frac{d}{d\xi}\omega(\xi) + \left((k_2^2 - k_1^2)/\alpha^2 \right)\omega(\xi) = 0, $$ so the same thing but without the ##- ik_2/\alpha \frac{(1-\xi)}{\xi}\omega(\xi)## term. So he gets an hypergeometric equation and I don't. I don't know where my mistakes are.

Any help would be much appreciated.

Thanks !

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