Let us assume a stream of non-relativistic electrons of kinetic energy

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Let us assume a stream of non-relativistic electrons of kinetic energy E travelling along the x-axis experiences an abrupt change in potential from 0 to V1 at x = 0, where E > V1. At x = a, the electrons experience another step change in potential from V1 to V2, as shown in the sketch below. (a) Starting from the "time independent" Schrödinger equation for the particle wavefunctions in different regions (i. e. x < 0, 0 < x < a and x > a), write down the general solutions for the allowed eigenfunctions. (b) Draw a suitably labelled diagram to appropriately represent the eigenfunctions in all regions. (c) Using appropriate boundary conditions to ensure the eigenfunctions are well behaved, show that there will be 100% transmission of electrons to the region x > a when V2 = 0 and that the wavelength of the eigenfunction in the region 0 < x < a is 2a.

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Let us assume a stream of non-relativistic electrons of kinetic energy $E$ travelling along the $x$ -axis experiences an abrupt change in potential from 0 to $V_{1}$ at $x = 0$ , where $E > V_{1}$ . At $x = a$ , the electrons experience another step change in potential from $V_{1}$ to $V_{2}$ , as shown in the sketch below. (a) Starting from the "time independent" Schrödinger equation for the particle wavefunctions in different regions (i.e. $x < 0, 0 < x < a$ and $x > a$ ), write down the general solutions for the allowed eigenfunctions. (b) Draw a suitably labelled diagram to appropriately represent the eigenfunctions in all regions. (c) Using appropriate boundary conditions to ensure the eigenfunctions are well behaved, show that there will be $100 %$ transmission of electrons to the region $x > a$ when $V_{2} = 0$ and that the wavelength of the eigenfunction in the region $0 < x < a$ is $2 a$ .