To describe the phenomenon of a particle failing to tunnel through a finite potential well, we can use the Schrödinger equation, which is the fundamental equation of quantum mechanics. The Schrödinger equation describes the behavior of a quantum particle in terms of its wave function.
The time-independent Schrödinger equation for a particle of mass mmm in a one-dimensional potential well is given by:
H^ψ(x)=Eψ(x)hat{H} psi(x) = E psi(x)H^ψ(x)=Eψ(x),
where H^hat{H}H^ is the Hamiltonian operator, ψ(x)psi(x)ψ(x) is the wave function of the particle as a function of position xxx, EEE is the energy of the particle, and ℏ is the reduced Planck's constant.
Inside the finite potential well region, the potential energy V(x)V(x)V(x) is non-zero, while outside the well it is higher. Therefore, we can divide the problem into three regions: the left region (Region I) where the potential is higher, the finite potential well region (Region II), and the right region (Region III) where the potential is higher again.
The wave function and its derivatives in each region can be written as:
Region I (left): ψI(x)=Aeik1x+Be−ik1xpsi_I(x) = A e^{ik_1x} + B e^{ -ik_1x}ψI(x)=Aeik1x+Be−ik1x,
Region II (well): ψII(x)=Ceik2x+De−ik2xpsi_{II}(x) = C e^{ik_2x} + D e^{ -ik_2x}ψII<