91Ó°ÊÓ

In quantum mechanics, continuity of the wave function is a key principle. At a boundary, the wave function on one side must smoothly transition to the wave function on the other side. This ensures that there is no sudden jump in the probability of finding the quantum particle, which would be unphysical. For the potential step problem, we have two regions defined by the wave functions: one for \( x < 0 \) and another for \( x > 0 \).

• In the region \( x < 0 \), \( \psi(x) = A e^{i k_1 x} + B e^{-i k_1 x} \).
• In the region \( x > 0 \), \( \psi(x) = C e^{i k_2 x} \).

At the boundary \( x = 0 \), the continuity condition \( \psi(0^-) = \psi(0^+) \) translates to:\[ A + B = C \]This equation shows how the wave functions from both sides must be equal exactly at the boundary. This requirement helps us solve for the variables \( B \) and \( C \) in terms of the given constants.

A potential step is a situation in quantum mechanics where the potential energy changes suddenly at a certain point. This kind of setup is ideal for exploring concepts such as reflection and transmission of quantum particles. In our exercise, this potential step occurs at \( x = 0 \).

• For \( x < 0 \), the potential \( U(x) = 0 \), meaning the particle moves in a region where there is no additional potential energy.
• For \( x > 0 \), the potential \( U(x) = U_0 < E \), introducing a step in potential energy lower than the particle's total energy \( E \).

This introduces a boundary condition that affects how the quantum particle behaves when moving between different potential regions. The potential step creates a boundary that must be crossed, causing some of the quantum particles to reflect, while others are transmitted.

Reflection in quantum mechanics is a fascinating phenomenon. When a quantum particle encounters a potential step, part of its wave function is reflected back. This is akin to a light wave reflecting off a mirror. In our scenario, the reflection is represented by the term \( B e^{-i k_1 x} \) for \( x < 0 \). The reflection coefficient \( B \) quantifies how much of the particle's wave function is reflected due to the potential step.

• Reflection depends on the difference in potential and the particle's energy.
• For a step with \( U_0 < E \), not all particles are reflected.

The derived expression for \( B \) is:\[ B = \frac{k_1 - k_2}{k_1 + k_2} A \]This shows how the portion of the wave function that is reflected depends on the wave numbers \( k_1 \) and \( k_2 \), and hence on the energies and momenta of the particles on either side of the step.

Transmission occurs when a quantum particle crosses a potential step. Unlike classical particles which would only pass over a barrier if they have enough energy, quantum particles have a probability of being transmitted even at lower energies, a hallmark of quantum tunneling. In this problem, the term \( C e^{i k_2 x} \) for \( x > 0 \) represents the transmitted wave function. The transmission coefficient \( C \) indicates how much of the particle is transmitted through the potential step.

• Transmission relates to the continuity of the wave function and its derivative.
• Even when the potential energy is less than total energy, transmission can occur.

The formula for \( C \) is:\[ C = \frac{2k_1}{k_1 + k_2} A \]This tells us how the wave function adjusting itself to the step leads to a transmitted portion, determined by the relative wave numbers. It captures the essence of quantum mechanics where particles exhibit wave-like behavior, allowing them to be both reflected and transmitted at potential boundaries.