Question

Equation \((5-16)\) gives infinite well energies. Because equation \((5-22)\) cannot be solved in closed form, there is no similar compact formula for finite well energies. Still, many conclusions can be drawn without one. Argue on largely qualitative grounds that if the walls of a finite well are moved closer together but not changed in height, then the well must progressively hold fewer bound states. (Make a clear connection between the width of the well and the height of the walls.)

Short answer

In a quantum well, bound states correspond to the energy levels that lie below the energy at the boundaries of the well. If the width of the finite well is decreased without changing the well depth, the available 'space' for the particle to 'fit' within the well's potential energy is reduced, leading to fewer potential energy levels and hence fewer bound states.

Step-by-step solution

Explanation of Bound States

In quantum mechanics, a particle is in a bound state if its energy is less than the potential energy at the boundaries. In this context, the potential energy at the boundaries is defined by the depth (or height) of the well and the boundaries themselves — the width of the well.

Correlation between Well Width and Bound States

When the width of the well is decreased (i.e., the walls are moved closer together), the 'space' within the potential energy of the well for the particle to 'fit' is reduced. This phenomenon results in a decreased number of potential energy levels below the energy of the well itself.

Linking Width to Number of Bound States

As a consequence of the reduction in well width detailed in Step 2, there will be fewer energy states below the height of the well's walls that a particle can occupy. Thus fewer bound states are available to a particle.

Key concepts

Bound States

Bound states in quantum mechanics refer to the condition where a particle is confined to a particular region in space because its energy is lower than the potential energy outside this region. Think of a marble in a bowl. The marble cannot escape unless it accumulates enough energy to get over the edge of the bowl. Similarly, in a quantum system, a particle is "trapped" in a region if its energy is less than the potential barriers surrounding it.

• Bound states are crucial in determining how particles behave in quantum systems.
• They only exist when the particle’s energy is insufficient to overcome the potential barrier, confining the particle within the boundary.
• The particle's wavefunction tends to diminish outside these boundaries, suggesting confinement.

Understanding bound states is key to solving many quantum mechanical problems, as it defines conditions where particles remain within finite regions.

Finite Potential Well

A finite potential well is a simplified model used in quantum mechanics to understand the behavior of particles within confined boundaries. Unlike an infinite well, where walls are impenetrable, a finite potential well has boundaries that particles can, in theory, penetrate if they possess sufficient energy.
When analyzing a finite potential well, the energy levels are not predefined as in an infinite well. Instead:

• Particles have a probability of existing both inside and slightly outside the well, thanks to tunneling effects inherent to quantum systems.
• The depth and width of the well determine the potential energy landscape and possible bound states that can exist.
• Calculating exact energy levels can be complex, often relying on numerical methods due to the absence of simple formulas unlike the infinite well scenario.

This concept acts as a building block for understanding more advanced quantum mechanical systems and applications.

Energy Levels

Energy levels within quantum mechanical systems are discrete values of energy that particles can occupy. These levels arise because of the boundary conditions imposed by the system, such as the confines of a finite potential well.

• The separation of these energy levels is influenced by the size and shape of the confining potential well.
• As well width decreases, energy differences between levels typically increase, indicating fewer available states.
• Calculated using wavefunctions, these energy levels provide insight into where and how particles might be found within a system.

This discreteness in energy levels is a hallmark of quantum mechanics, signifying that unlike classical systems, particles cannot possess arbitrary energies.

Particle in a Box

The 'particle in a box' model is one of the most fundamental problems tackled in quantum mechanics. It illustrates how quantum levels arise through confinement.

• In an infinite potential well, or 'box,' a particle is confined completely, unable to penetrate the walls, leading to quantized energy levels derived from simple formulas.
• The finite potential well version adds complexity, permitting the particle to have a non-zero probability of being outside, a feature not present in the infinite model.
• By adjusting the width of this "box," one can observe changes in the spacing and number of energy levels, echoing the fundamental principle that spatial confinement directly affects quantum behavior.

Understanding these models is crucial for comprehending quantum mechanical systems and interpreting the behavior of particles under different conditions.