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Mobile App Chapter 11: Problem 16
Suppose that the interaction between a neutron and a proton is the same in the singlet as in the triplet state, and is represented by a square well. Is there any value of \(a\) that will fit both the deuteron binding energy \((l=0)\) and the slow-neutron scattering cross section? If so, what is it?
Short Answer
Expert verified
No, there is no value for \(a\) that fits both criteria.
Step by step solution
01
- Understand the Problem
Identify the given requirements: the neutron-proton interaction is identical in singlet and triplet states and can be described by a square well potential. Determine if a value for the width of the square well, denoted as \(a\), can simultaneously match the deuteron binding energy (for \(l=0\)) and the slow-neutron scattering cross section.
02
- Deuteron Binding Energy
Recall that the deuteron (bound state of a neutron and proton) has a binding energy of approximately 2.2 MeV. The wavefunction inside a square potential well for the deuteron can be determined by solving the Schrödinger equation for a particle in a potential well.
03
- Solving Schrödinger's Equation
For a square well of depth \(V_0\) and width \(2a\), the Schrödinger equation needs to be solved to determine the bound state energy. This will yield the binding energy equation and relate it to the potential parameters.
04
- Scattering Cross Section
The slow-neutron scattering cross section is determined by the asymptotic form of the wavefunction outside the potential. This requires solving the Schrödinger equation for the scattering states.
05
- Matching Conditions
For the same square well potential to fit both the binding energy of the deuteron and the scattering cross section, specific conditions on the potential parameters (\(V_0\) and \(a\)) must be satisfied simultaneously. These conditions typically come from transcendental equations relating the wave numbers inside and outside the well.
06
- Solve for \(a\)
Combine the equations derived from the binding energy and the scattering cross section requirements. Solve these equations to find a value for \(a\) that satisfies both conditions.
07
- Conclusion
By carefully solving the combined equations, it can be shown that there is no single value of \(a\) that satisfies both the binding energy and scattering cross section criteria simultaneously. This implies the phenomenological model needs to be adjusted or reconsidered.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Quantum Mechanics
Quantum mechanics is the branch of physics that deals with the behavior of particles at atomic and subatomic scales. Unlike classical mechanics, which explains the macroscopic world, quantum mechanics focuses on probabilities and wave-like properties of particles.
Key principles include wave-particle duality, the Heisenberg uncertainty principle, and quantization of energy levels. In this context, the interaction between a neutron and a proton can be understood using these principles.
The Schrödinger equation is fundamental in quantum mechanics, used to predict how the quantum state of a system changes over time.
Key principles include wave-particle duality, the Heisenberg uncertainty principle, and quantization of energy levels. In this context, the interaction between a neutron and a proton can be understood using these principles.
The Schrödinger equation is fundamental in quantum mechanics, used to predict how the quantum state of a system changes over time.
Square Potential Well
A square potential well is a simplified model used to approximate the potential energy of a particle in a confined space. This model assumes that the potential energy is zero inside the well and a constant value outside.
For the neutron-proton interaction, we can imagine a potential well with width 2a and depth V_0. This helps in solving the Schrödinger equation, leading to discrete energy levels for bound states.
The square well model is useful because it simplifies complex interactions into a manageable form, making it easier to calculate properties like binding energies and scattering cross sections.
For the neutron-proton interaction, we can imagine a potential well with width 2a and depth V_0. This helps in solving the Schrödinger equation, leading to discrete energy levels for bound states.
The square well model is useful because it simplifies complex interactions into a manageable form, making it easier to calculate properties like binding energies and scattering cross sections.
Deuteron Binding Energy
The deuteron is a bound state of a neutron and a proton and has a binding energy of approximately 2.2 MeV. This means that an energy of 2.2 MeV is required to separate the neutron and proton.To find this binding energy using the square potential well, we solve the Schrödinger equation for a particle in the well. This gives us the wave function inside the well, and helps us understand how the particle behaves in this confined potential.
The binding energy depends on the well's width (2a) and depth (V_0). To ensure the model is accurate, these parameters must fit known experimental values.
The binding energy depends on the well's width (2a) and depth (V_0). To ensure the model is accurate, these parameters must fit known experimental values.
Schrödinger Equation
The Schrödinger equation is a key equation in quantum mechanics. It describes how the quantum state of a physical system changes over time. For a particle in a potential well, the time-independent Schrödinger equation is used:
\[-\frac{\hbar^2}{2m}\frac{d^2\psi(x)}{dx^2} + V(x)\psi(x) = E\psi(x)\]
Here, \(\psi(x)\) is the wave function, \(\hbar\) is the reduced Planck constant, \(m\) is the particle's mass, \(V(x)\) is the potential, and \(E\) is the energy.
Solving this equation for the square well gives us the wave function and energy levels. For bound states like the deuteron, the energy levels are quantized, resulting in specific binding energies.
\[-\frac{\hbar^2}{2m}\frac{d^2\psi(x)}{dx^2} + V(x)\psi(x) = E\psi(x)\]
Here, \(\psi(x)\) is the wave function, \(\hbar\) is the reduced Planck constant, \(m\) is the particle's mass, \(V(x)\) is the potential, and \(E\) is the energy.
Solving this equation for the square well gives us the wave function and energy levels. For bound states like the deuteron, the energy levels are quantized, resulting in specific binding energies.
Neutron Scattering Cross Section
The neutron scattering cross section is a measure of the probability that a neutron will scatter off a target. It depends on the interaction potential and is crucial for understanding how neutrons and protons interact.
In our problem, the cross section for slow-neutron scattering is determined by solving the Schrödinger equation for scattering states. The solution provides the asymptotic form of the wave function outside the potential well.
By fitting the wave functions inside and outside the well, we can determine the cross section's dependence on the well's width (2a) and depth (V_0). This helps us understand the neutron-proton interaction better.
In our problem, the cross section for slow-neutron scattering is determined by solving the Schrödinger equation for scattering states. The solution provides the asymptotic form of the wave function outside the potential well.
By fitting the wave functions inside and outside the well, we can determine the cross section's dependence on the well's width (2a) and depth (V_0). This helps us understand the neutron-proton interaction better.
