Question

Sketch the two lowest energy wave functions for an electron in an infinite potential well that is \(20 \mathrm{nm}\) wide and a finite potential well that is \(1 \mathrm{eV}\) deep and is also \(20 \mathrm{nm}\) wide. Using your sketches, can you determine whether the energy levels in the finite potential well will be lower, the same, or higher than in the infinite potential well?

Short answer

Answer: The energy levels in the finite potential well are lower than in the infinite potential well.

Step-by-step solution

Analyze infinite potential well

An infinite potential well has zero potential energy (V(x)=0) within the well and infinite potential energy outside the well. The width of the well is given as 20 nm. The Schrödinger equation becomes a one-dimensional problem, with the boundary condition of zero probability outside the well. The general solutions of the wave function inside the well are of the form: $$\psi_n(x) = A \sin(k_n x)$$ where \(k_n = \frac{n \pi}{L}\), and \(L\) is the width of the well (20 nm).

Sketch lowest energy wave functions for infinite potential well

For an infinite potential well, the two lowest energy wave functions are given by: $$\psi_1(x) = A \sin\left(\frac{\pi x}{L}\right)$$ $$\psi_2(x) = A \sin\left(\frac{2\pi x}{L}\right)$$ We can sketch the wave functions inside the well between 0 and 20 nm. The first wave function, \(\psi_1(x)\), is a single sine wave. The second wave function, \(\psi_2(x)\), is two sine waves within the well.

Analyze finite potential well

A finite potential well has zero potential energy within the well (V(x)=0) and a finite potential energy (\(V_0 = 1\) eV) outside the well. The Schrödinger equation becomes again a one-dimensional problem. The width of the well is given as 20 nm. Inside the well, the wave function has the same form as in the infinite potential well. Outside the well, the wave function is a decaying exponential: $$\psi(x) = B e^{-(x-L)/a}$$ where \(a = \frac{\hbar^2}{2m_eV_0}\).

Sketch lowest energy wave functions for finite potential well

For a finite potential well, the two lowest energy wave functions are also given by: $$\psi_1(x) = A \sin\left(\frac{\pi x}{L}\right)$$ $$\psi_2(x) = A \sin\left(\frac{2\pi x}{L}\right)$$ Inside the well, the wave functions have the same shape as those in the infinite potential well. However, outside the well, the wave functions are decaying exponentials. We can sketch the wave functions, making sure to include the exponential decay outside the well.

Compare energy levels between finite and infinite potential wells

The energy levels for finite and infinite potential wells can be compared by looking at the wave functions. For the infinite potential well, the energy levels are quantized and given by: $$E_n = \frac{n^2\pi^2\hbar^2}{2m_eL^2}$$ For the finite potential well, energy levels are also quantized but are affected by the exponential decay of the wave functions outside the well. The quantization is less strict, and the energy levels will be slightly lower than those in the infinite potential well due to the penetration of the wave function into the barrier. In conclusion, the energy levels in the finite potential well will be lower than in the infinite potential well.

Key concepts

Infinite Potential Well

The infinite potential well is a cornerstone concept in quantum mechanics. Imagine it as a box with perfectly rigid walls. Within this box, potential energy is zero, while outside, it is infinite. This means particles inside cannot escape.

• Potential inside the well: 0
• Potential outside the well: Infinite
• Width of the well: 20 nm

In this scenario, a particle’s wave function must vanish at the walls, leading to specific allowed energy states that are quantized. The solutions to the Schrödinger equation here are sine waves confined strictly within the well. These wave functions are described as \( \psi_n(x) = A \sin(\frac{n \pi x}{L}) \), where \( n \) is a positive integer. Thus, only standing waves that fit perfectly within the box are possible.

Finite Potential Well

A finite potential well offers a more realistic scenario compared to the infinite well. Here, the potential energy inside the well is still zero, but outside, it is finite, say \( 1 \, \text{eV} \). This reflects situations where a particle might have a small chance of existing outside the well due to the finite potential barrier.

• Potential inside the well: 0
• Potential outside the well: Finite (1 eV)
• Width of the well: 20 nm

The wave function within the well is similar to that of the infinite well, but decays exponentially outside. This means particles have a nonzero probability of tunneling through the walls. Outside the well, the wave function takes the form \( \psi(x) = B e^{-(x-L)/a} \), where \( a \) depends on factors like the particle’s mass \( m_e \) and the barrier height \( V_0 \). This tunneling effect leads to energy levels being slightly lower than in the infinite case.

Schrödinger Equation

The Schrödinger equation forms the backbone of quantum mechanics and describes how quantum systems evolve over time. In the context of potential wells, it becomes a differential equation used to solve for the particle's wave function.

• Time-independent Schrödinger Equation: \( -\frac{\hbar^2}{2m} \frac{d^2\psi}{dx^2} + V(x)\psi = E\psi \)

Within an infinite potential well, we have \( V(x) = 0 \) leading to a simpler form: \[-\frac{\hbar^2}{2m} \frac{d^2\psi}{dx^2} = E\psi \] This equation's solutions inside an infinite or finite well vary based on boundary conditions: zero probability outside for infinite wells, and exponential decay for finite wells.

Energy Levels

In quantum mechanics, the concept of energy levels indicates that a system can only exist in certain discrete energy states. Each energy level corresponds to a different wave function. For infinite potential wells, energy levels are given by:\[ E_n = \frac{n^2\pi^2\hbar^2}{2m_eL^2} \]

• The energy is quantized in terms of integer \( n \)
• Larger \( n \) imply higher energy levels

For finite wells, the concept remains, but the energy levels are slightly altered due to the wave function’s overlap into the classically forbidden region outside the well. This tunneling effect effectively results in lower energy levels than those in an infinite potential well.

Wave Functions

Wave functions are critical in quantum mechanics as they provide information about the probability of finding a particle in a given space. In potential wells, the form and properties of these functions are dictated by the Schrödinger equation and boundary conditions.

• Inside well: Typically sine or cosine for bound states
• Outside well (finite case): Exponentially decaying functions

For an infinite potential well, wave functions are like "loops" fitting only within the well, ensuring viability only at distinct energy states. In contrast, finite well wave functions "reach out" beyond the well, showcasing the quantum mechanical phenomenon of tunneling, allowing for deeper insights into energy state behavior.