Question

The deeper the finite well, the more states it holds. In faet, a new state, the \(n t h\), is added when the well's depth be the case based only on \(k=\sqrt{2 m E / \hbar^{2}}\), the shape of the wave inside. and the degree of penetration of the classically forbidden region expected for a state whoes energy \(E\) is only negligibly below \(U_{0}\) ( (b) How many states would be found up to this same "height" in an infinite well.

Short answer

The exact number of states in finite and infinite wells depends on the specific values of parameters given, and the penetration depth under consideration. But as a general principle, an infinite well will hold more states than a finite one for the same energy level.

Step-by-step solution

Analyze the State of a Finite Well

Firstly, it's important to understand that the number of quantum states in a finite well is related to the well's depth. It's given that \(k=\sqrt{2 m E / \hbar^{2}}\) is the essential factor that affects the number of states. So, calculate the wave number for a given energy \(E\) just below \(U_{0}\).

Calculate the Penetration of the Classically Forbidden Region

Next, it's necessary to estimate the degree of penetration into the classically forbidden region of the well to find the state with energy \(E\). The penetration depth of the wave function into the classically forbidden region affects the spacing of the energy levels and hence the number of bound states in the well.

Calculate Number of States in an Infinite Well

Lastly, calculate the number of states in an infinite well up to the same 'height', or energy \(E\). In an infinite well, the total number of states up to energy \(E\) is generally more than in a finite well because there is no classically forbidden region limiting the number of bound states.

Key concepts

Finite Potential Well

The finite potential well is a fundamental concept in quantum mechanics. It refers to a potential energy landscape where the potential energy is lower inside a specific region—known as the 'well'—and higher outside it. Because the well is "finite," the particle can have bound states where its energy level is less than the potential outside the well, but more than zero.

Within a finite potential well, the number of quantum states depends on several factors, notably the depth of the well. The deeper the well, the more energy levels or quantum states it can hold. The energy levels in this type of potential are quantized, meaning they can only take on specific values.

The term "quantum states" refers to the distinct energy levels that an electron can occupy within the well. To find how many states there are, we often use the wave number formula: \[ k = \sqrt{\frac{2mE}{\hbar^2}}\]where \( k \) is the wave number, \( m \) is the particle's mass, \( E \) is its energy, and \( \hbar \) is the reduced Planck's constant. The number of quantum states can increase as the well becomes deeper.

Quantum States

Quantum states are the different levels within a potential well that an electron can occupy when its energy is less than the potential at the finite well's boundary. In a finite potential well, these states are quantized, implying that an electron can only exist at certain distinct energy levels and not in between.

This quantization of states comes from the nature of wave-particle duality, where particles like electrons exhibit both particle and wave characteristics. As a result, not every energy value is possible within the well, leading to these discrete levels.
Quantum mechanics provides the tools to predict these states by solving the Schrödinger equation for the potential well. By knowing the potential depth and width of the well, one can calculate the possible energy levels and understand how many states the well can support. The relation between the number of states and well depth is direct—the deeper the well, the more states it can accommodate, creating a richer spectrum of possible quantum behavior.

Classically Forbidden Region

In quantum mechanics, the classically forbidden region refers to a region where, traditionally under classical mechanics, a particle would not be expected to be found due to insufficient energy. However, due to quantum effects, there is a probability of a particle being present in such regions. This is unlike classical systems where a particle confined in a potential well cannot exist in a region beyond its total energy.

In the context of a finite potential well, the classically forbidden region exists just outside the well. This is where the wave function does not vanish but diminishes exponentially, indicating that the particle has "tunneled" into an energy region it would not classically appear. Funnily enough, it's this tunneling effect that shows the stark difference between quantum mechanics and classical physics.
The penetration depth is the measure of how far the wave function extends into this forbidden region, and it affects the number of quantum states within the finite well. This phenomenon exemplifies one of the most distinctive features of quantum mechanics, showing how particles behave in ways unanticipated by classic theories.

Infinite Potential Well

The infinite potential well is an idealized model in quantum mechanics where the potential outside a region is considered infinitely large, effectively trapping particles inside. Unlike the finite well, where the potential is finite just outside the well, an infinite well has vertical walls, representing non-penetrable boundaries.

In an infinite potential well, the number of quantum states is different compared to a finite well. Here, the number of energy levels can be easily calculated because there's no chance for the particle's wave function to extend beyond the boundary due to the infinite potential.
The absence of the classically forbidden region in an infinite potential well means more closely spaced states can be packed into the same energy range compared to a finite well. The lack of penetration allows for clear, distinct quantization. The energy levels are purely determined by the size of the well and the mass of the particle. Thus, infinite wells are often used as a simplified model that approximates certain conditions where boundary effects of a finite well are negligible. By comparing both finite and infinite wells, we can better understand quantum confinement and the impact boundaries have on quantum states.