Answer to Question #164734 in Quantum Mechanics for Mohammed Ismail

Question #164734

find Approximation transmission probability in quantum mechanics E<U


1
Expert's answer
2021-02-19T18:58:44-0500

Answer

Answer

Let for electron has wave function

ψeikx\psi\propto e^\frac{ikx}{\hbar}

The effective momentum of electron is given by


2k22m=qϕ\frac{\hbar^2 k^2}{2m*}=-q\phi

k=i2mqϕ2k=i\sqrt{\frac{2m*q\phi}{\hbar^2}}

So wavefunction become

ψei2mqϕ2x\psi\propto e^{i\sqrt{\frac{2m*q\phi}{\hbar^2}}x}


ψ(x=d)ψ(x=0)e2mqϕ2.de2mqϕ2.0e2mqϕ2d\frac{\psi (x=d) }{\psi(x=0) }\propto \frac{ e^{-\sqrt{\frac{2m*q\phi}{\hbar^2}}.d}}{e^{-\sqrt{\frac{2m*q\phi}{\hbar^2}}.0}}\propto e^{-\sqrt{\frac{2m*q\phi}{\hbar^2}}d}


Let JI

be the incident current and JT

be the transmitted current. Then


Transmission probability


TJTJ0(ψ(x=d)ψ(x=0))2e22mqϕ2dT\propto \frac{J_T}{J_0}\propto (\frac{\psi(x=d) }{\psi(x=0) }) ^2\\\propto e^{-2\sqrt{-\frac{2m*q\phi}{\hbar^2}}d}


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