C A particle is confined to a finite box of length L. In the n^th state, the wave function has

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C A particle is confined to a finite box of length L. In the...

C A particle is confined to a finite box of length $L$. In the $n^{\text {th }}$ state, the wave function has $n-1$ nodes. The wave function must make a smooth transition from sinusoidal inside the box to a decaying exponential outside-there can't be a kink at the wall. (a) Make some sketches to show that the wavelength $\lambda_{n}$ inside the box must fall in the range $2 L / n<\lambda_{n}<2 L(n-1)$. (b) Show that the energy levels $E_{n}$ in the finite box satisfy $$ (n-1)^{2} E_{1}<E_{n}<n^{2} E_{1}$$ where $E_{1}=h^{2} /\left(8 m L^{2}\right)$ is the ground-state energy for a box of length $L$ with infinite potential energy outside the box.

C A particle is confined to a finite box of length $L$. In the $n^{\text {th }}$ state, the wave function has $n-1$ nodes. The wave function must make a smooth transition from sinusoidal inside the box to a decaying exponential outside-there can't be a kink at the wall.
(a) Make some sketches to show that the wavelength $\lambda_{n}$ inside the box must fall in the range $2 L / n<\lambda_{n}<2 L(n-1)$. (b) Show that the energy levels $E_{n}$ in the finite box satisfy $$
(n-1)^{2} E_{1}<E_{n}<n^{2} E_{1}$$ where $E_{1}=h^{2} /\left(8 m L^{2}\right)$ is the ground-state energy for a box of length $L$ with infinite potential energy outside the box.

Key Concepts

Energy Quantization and Level Spacing

The quantization of energy in a confined system arises from the constraints imposed by the allowable wave functions. In an infinite potential well, the energy levels scale exactly with the square of the integer quantum number. However, in a finite well, the energy levels still exhibit a dependence on the quantum number but with bounds that reflect the altered spatial behavior of the wave function near the boundaries. This results in energy levels that lie within specific ranges relative to the ground state energy, capturing the effects of both the confined oscillatory behavior and the exponential decay outside the well.

Wavelength Constraints in Confined States

For a particle confined in a finite well, the wavelength of the wave function inside the well is not arbitrary but must satisfy certain inequalities due to the boundary conditions and number of nodes. This means that the wavelength must be within a specific range relative to the size of the well and the quantum number of the state. This concept is analogous to the standing wave conditions seen in simpler models but adjusted for the finite potential barrier.

Boundary Conditions and Wave Function Smoothness

The requirement that the wave function and its first derivative be continuous at the boundaries of the potential well is fundamental in quantum mechanics. This smooth matching condition, especially at the border between the oscillatory region inside the well and the exponential decay outside, restricts the allowable forms of the wave function. It imposes constraints on the wavelength of the particle’s state within the well and plays a crucial role in determining the energy quantization.

Finite Potential Well

In quantum mechanics, a finite potential well refers to a region of space where the potential energy is lower than in the surrounding regions but is not infinitely large outside. This leads to bound states where the particle's wave function is oscillatory inside the well and decays exponentially outside. The finite nature of the well introduces additional complexities compared to the infinite potential well, particularly in how the wave function behaves at the boundaries.

Wave Function Nodes and Quantization

The number of nodes (points where the wave function is zero) in a bound state is directly related to the quantum number associated with that state. Each additional node corresponds to a higher state with increased energy. In this context, the number of nodes affects the spatial oscillations of the wave function within the well, and the quantization condition emerges from the requirement that the wave function fits within the well in a way that complies with the boundary conditions.

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