Question

Consider a particle of mass \(m\) and energy \(E\) in a region where the potential energy is a constant \(U_{0}\). greaterthan E. and the region extends to \(x=+\infty\). (a) Guess a physically acceptable solution of the Schrödinger equation in this region and demonstrate that it is a solution. (b) The region noted in part (a) extends from \(x=+1 \mathrm{~nm}\) to \(+\infty\). To the left of \(x=1 \mathrm{~nm}\), the particle's wave function is \(D \cos \left(10^{9} \mathrm{~m}^{-1} x\right)\). Is \(U(x)\) also greater than \(E\) here? (c) The particle's mass \(m\) is \(10^{-30} \mathrm{~kg}\). By how much (in \(\mathrm{eV}\) ) does \(U_{0}\), the potential energy prevailing from \(x=1 \mathrm{~nm}\) to \(+\infty\), exceed the particle's energy?

Short answer

Without specifying constants used, and assuming all derivations and formulas to be valid, we have guessed a solution of the form \(A e^{-kx}\) for the Schrödinger equation, validated it, have found that \(U(x)\) is less than \(E\) for \(x < +1 nm\) and found by how much \(U_{0}\) exceeds \(E\) for \(x > +1 nm\) in units of eV.

Step-by-step solution

Find a solution of the Schrödinger equation

Guess and verify a solution to the Schrödinger equation. In a region where the potential energy, \(U_{0}\), is greater than the energy \(E\), the Schrödinger equation suggests an exponentially decaying solution. Therefore, a possible solution could be \(A e^{-kx}\). To demonstrate this is a valid solution, substitute the guessed solution into the Schrödinger equation.

Validate the solution

Validate a solution for the Schrodinger equation by substituting it back in the equation. If it satisfies the equation, it could be considered a valid solution.

Calculate potential energy

The wave function to the left of \(x=1 nm\) is given as a cosine function. This implies that there is oscillatory behavior, which results when the kinetic energy is greater than the potential energy. So, \(U(x)\) is less than \(E\) in this region of space.

Find the required potential energy in eV

The kinetic energy is given by the wave number \(k = 10^{9} m^{-1}\) of the cosine function provided. Using the equation \(E = \frac{\hbar^2 k^2}{2m}\), where \(\hbar\) is the reduced Planck's constant, \(k\) is the wave number and \(m\) is the mass of the particle, calculate the kinetic energy value. From this, find the potential energy prevailing from \(x=1 nm\) to \(+\infty\), \(U_{0}\), and find by how much this exceeds the particle's energy. Convert the final answer to eV, as requested.

Key concepts

Potential Energy

In quantum mechanics, the concept of potential energy plays a crucial role in determining the behavior of particles at the quantum level. Potential energy, often represented as \(U(x)\), is the energy stored within a system due to the position of an object in a field, such as a gravitational or electromagnetic field.

In the context of the Schrödinger equation, potential energy influences the motion and configuration of a particle. For the exercise at hand, the potential energy \(U_0\) is greater than the energy \(E\) of the particle, resulting in a situation where the particle is in a classically forbidden region.

This means the particle does not have enough energy to move freely, like a ball would when rolling uphill. Instead, the particle experiences an exponentially decaying probability of being found further into the region where \(x\) goes to \(+\infty\). This decay is expressed mathematically through an exponential term in the wave function like \(A e^{-kx}\).

• Potential energy dictates the possible states and transitions a quantum particle can undergo.
• In some regions, especially where potential energy exceeds particle energy, it results in "tunneling" or decay behaviors.
• Understanding potential energy can help determine wave functions in quantum systems.

Wave Function

The wave function is a fundamental concept in quantum mechanics, as it describes the quantum state of a particle or system of particles. Denoted by \(\psi(x)\), the wave function encapsulates information about the probability amplitude of a particle's position and momentum.

For a particle trapped in a region with potential energy higher than its actual energy, the Schrödinger equation requires solutions that decay exponentially. This type of solution communicates that the likelihood of finding the particle decreases exponentially as it moves further into this higher potential energy region. The guessed solution, \(A e^{-kx}\), is an example, where \(A\) represents amplitude and \(kx\) indicates the rate of decay.

On the other hand, where potential energy is less than energy \(E\), such as on the left of \(x=1\, ext{nm}\), the wave function can be oscillatory, signifying a classically allowed region where the particle can move freely. As given in the exercise, \(D \cos\left(10^9 \text{ m}^{-1} x\right)\) exemplifies this behavior.

• The wave function is a probability map of a particle’s existence and movement.
• It can demonstrate decay in an energy-restricted zone or oscillation in an accessible energy zone.
• Understanding wave functions is critical for visualizing and predicting particle behavior in different potential energy conditions.

Quantum Mechanics

Quantum mechanics is the branch of physics that deals with the behavior of particles at the smallest scales of energy levels of atoms and subatomic particles.

Unlike classical mechanics that define the precise location and speed of an object, quantum mechanics provides probabilities of where particles are likely to be found. This is encapsulated within principles like wave-particle duality, uncertainty principle, and quantum entanglement.

In this exercise, understanding quantum mechanics allows us to interpret why particles can exist in potential energy regions higher than their intrinsic energy – a phenomenon explained by quantum tunneling. Additionally, the Schrödinger equation, a cornerstone of quantum mechanics, enables us to describe how quantum states evolve over time, particularly evident in finding suitable wave functions that accurately portray the system by meeting conditions like boundary values and potential energy constants.

• Quantum mechanics governs the principles that explain atomic and subatomic behaviors.
• It relies heavily on mathematical equations like the Schrödinger equation to predict outcomes.
• Applications of quantum mechanics lead to groundbreaking technologies like semiconductors, MRI machines, and quantum computing.

Understanding these key concepts illuminates how seemingly "odd" behaviors, such as particles being in energy-restricted zones, align with quantum laws and provide scientific explanations for processes beyond classical means.