Infinite potential well with a step barrier
In the realm of quantum mechanics, the study of potential wells provides crucial insights into particle behavior under quantum confinement. Specifically, the one-dimensional infinite potential well with a step barrier presents a rich field for theoretical exploration. Here, I analyze a scenario where a particle is confined within such a system, illustrating both the physical setup and mathematical formulation necessary to understand the underlying quantum dynamics.
The potential V(x) for this system is defined as:
V(x) = \begin{cases} \infty & \quad x \le 0 \\ 0 & \quad 0 \le x \le L \\ V_0 & \quad x \ge L \end{cases}
This setup introduces an infinite barrier at x \le 0 and a step potential of height V_0 at x \ge L. The most interesting aspect arises when considering the first eigenstate energy E_1 = \frac{V_0}{2}, which lies precisely at half the potential step, thus providing a critical insight into energy quantization and confinement.
Quantum state within the well
Within the well, the potential V(x) = 0, leading to the simplified one-dimensional Schrödinger equation:
-\frac{\hbar^2}{2m} \frac{\mathrm d^2\psi(x)}{\mathrm d x^2} = E\psi(x)
The solutions to this equation are a combination of sine and cosine functions. Given the boundary condition at x = 0 due to the infinite potential barrier, the wave function \psi_W(x) simplifies to:
\psi_W(x) = A\sin(kx), \quad k = \sqrt{\frac{2Em}{\hbar^2}}
Transition Across the Barrier
As the potential rises to V_0 beyond x = L, the Schrödinger equation modifies to account for the potential difference, which, for the case where E_1 = \frac{V_0}{2}, yields solutions involving exponentially decaying functions:
\psi_B(x) = C,e^{-k (x-L)}, \quad k = \sqrt{\frac{2m\left(V_0 - E\right)}{\hbar^2}} = \sqrt{\frac{2mE}{\hbar^2}}
This provides a continuity at x = L, crucial for matching the wave functions and their derivatives at the boundary of the well and step barrier. This continuity is vital for ensuring physical and mathematical consistency of the wave function across the domain.
Mathematical Consistency and Physical Interpretation
To ensure continuity, I equate the wave functions and their first derivatives at x = L:
\begin{aligned} & A\,\sin(kL) = C \\ & kA\,\cos(kL) = -kC \\ \end{aligned}
These equations lead to conditions on kL that satisfy \tan(kL) = -1, which corresponds to kL = \frac{3}{4}\pi. This specific result ties the wave function’s spatial characteristics directly to the potential well’s physical dimensions and the particle’s energy state.
Normalization and Physical Reality
Normalizing the wave function, I calculate the coefficient A to ensure the total probability density across the space equals one. This involves integrating the square of the wave function across the entire space:
\begin{aligned} & A^2\left[\int_0^L \sin^2(kx) \, \mathrm dx + \int_L^\infty \frac{1}{2}e^{-2k (x-L)}\, \mathrm dx \right] = 1 \\ & A^2 \left[ \frac{L}{2} - \frac{\sin(2kL)}{4k} + \frac{1}{4k} \right] = 1 \end{aligned}
Using kL = \frac{3}{4}\pi, this leads to the normalization factor A, ensuring that the wave function correctly represents the probability density of finding the particle within specific regions of the system.
This analysis provides not just a deeper understanding of the quantum behavior of particles in constrained geometries but also aids in the design of quantum devices where such confinement effects are utilized.
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