Problem 74 - University Physics with Modern Physics

Question

Protons, neutrons, and many other particles are made of more fundamental particles called quarks and antiquarks (the antimatter equivalent of quarks). A quark and an antiquark can form a bound state with a variety of different energy levels, each of which corresponds to a different particle observed in the laboratory. As an example, the \(\psi\) particle is a low-energy bound state of a so-called charm quark and its antiquark, with a rest energy of 3097 MeV; the \(\psi(2 S)\) particle is an excited state of this same quark-antiquark combination, with a rest energy of 3686 MeV. A simplified representation of the potential energy of interaction between a quark and an antiquark is \(U(x)=A|x|,\) where \(A\) is a positive constant and \(x\) represents the distance between the quark and the antiquark. You can use the WKB approximation (see Challenge Problem 40.72 ) to determine the bound-state energy levels for this potential-energy function. In the WKB approximation, the energy levels are the solutions to the equation $$\int_{a}^{b} \sqrt{2 m[E-U(x)]} d x=\frac{n h}{2} \quad(n=1,2,3, \ldots)$$ Here \(E\) is the energy, \(U(x)\) is the potential-energy function, and \(x=a\) and \(x=b\) are the classical turning points (the points at which \(E\) is equal to the potential energy, so the Newtonian kinetic energy would be zero). (a) Determine the classical turning points for the potential \(U(x)=A|x|\) and for an energy \(E\) . (b) Carry out the above integral and show that the allowed energy levels in the WKB approximation are given by $$E_{n}=\frac{1}{2 m}\left(\frac{3 m A h}{4}\right)^{2 / 3} n^{2 / 3} \quad(n=1,2,3, \ldots)$$ (Hint: The integrand is even, so the integral from \(-x\) to \(x\) is equal to twice the integral from 0 to \(x\) . \((\mathrm{c})\) Does the difference in energy between successive levels increase, decrease, or remain the same as \(n\) increases? How does this compare to the behavior of the energy levels for the harmonic oscillator? For the particle in a box? Can you suggest a simple rule that relates the difference in energy between successive levels to the shape of the potential-energy function?

Step-by-Step Solution

Answer

The energy difference decreases as \(n\) increases, unlike the harmonic oscillator with constant energy differences.

Step 1: Determine Classical Turning Points

For the potential \(U(x) = A|x|\) and a given energy \(E\), the classical turning points occur where \(E = U(x)\). Therefore, \(E = A|x|\) implies the turning points are at \(x = \pm \frac{E}{A}\).

Step 2: Set Up the Integral in the WKB Approximation

The WKB approximation is given by \(\int_{a}^{b} \sqrt{2m[E - U(x)]} \, dx = \frac{nh}{2}\). Using symmetry, the integral from \(-x\) to \(x\) is twice the integral from 0 to \(x\). Thus, \[ \int_{-x}^{x} \sqrt{2m[E - A|x|]} \, dx = 2 \int_{0}^{x} \sqrt{2m[E - Ax]} \, dx. \]

Step 3: Calculate the Integral

Substitute \(U(x) = Ax\) and integrate: \[ \int_0^{x} \sqrt{2m(E - Ax)} \, dx = \int_0^{x} \sqrt{2mE - 2mAx} \, dx. \] Let \(u = (2mE - 2mAx)\), then \(du = -2mA \, dx\). Change of variables leads to \(\int \sqrt{u} \, \frac{du}{-2mA}\), which gives after integration, \(\frac{2}{3}\left(\frac{2mE}{A}\right)^{3/2}\).

Step 4: Apply the WKB Condition

The WKB quantization condition is \(2 \int_0^{\frac{E}{A}} \sqrt{2m[E - Ax]} \, dx = \frac{nh}{2}\). \[2 \left(\frac{2}{3}\left(\frac{2mE}{A}\right)^{3/2}\right) = \frac{nh}{2},\] leading to \(\left(\frac{2mE}{A}\right)^{3/2} = \frac{3nh}{4}\).

Step 5: Solve for Energy Levels

Solve the equation \(\left(\frac{2mE}{A}\right)^{3/2} = \frac{3nh}{4}\) for \(E\): \[E = \frac{1}{2m}\left(\frac{3mAh}{4}\right)^{2/3}n^{2/3}.\] Hence, the energy levels \(E_n\) can be found.

Step 6: Analyze Energy Level Differences

Observe the expression for \(E_n = \frac{1}{2m}\left(\frac{3mAh}{4}\right)^{2/3}n^{2/3}\). The difference \(E_{n+1} - E_n\) decreases as \(n\) increases because \(n^{2/3}\) grows slower than linearly. Unlike a harmonic oscillator where the difference is constant, a linear potential like this sees differences decrease as \(n\) increases.

Key Concepts

WKB approximation

To understand quark bound states, we use the WKB approximation, a handy tool for estimating energy levels in quantum mechanics when dealing with potentials that are smooth and slowly varying. This approximation simplifies complex quantum problems by providing a way to approximate the wave functions and energy levels of quantum systems.
In essence, the WKB approximation takes advantage of classical mechanics and converts it into a quantum problem. The core idea is to solve the Schrödinger equation approximately by using the integrals over the potential energy. When we talk about quark-antiquark pairs forming bound states, this approximation helps us to estimate the allowed energy levels by evaluating an integral involving the potential energy function and kinetic energy, represented as \[\int_{a}^{b} \sqrt{2m[E-U(x)]} \, dx = \frac{nh}{2}, \text{ where } n = 1, 2, 3, \ldots\]Here, \(E\) represents the energy of the bound state, \(U(x)\) is the potential energy function, \(m\) is the mass of the particles, \(h\) is Planck's constant, and \(a\) and \(b\) are the turning points where the particle changes direction.
This approximation is particularly useful in situations where an exact solution is difficult to obtain, making it a powerful tool for physicists studying systems like quark bound states.

energy levels

Energy levels in the context of quark bound states describe the specific energies a quark-antiquark system can occupy as a particle. When quarks form bound states, they do so at discrete energy values, much like the steps on a staircase. Each energy level corresponds to a particular state or configuration of the system.
The energy levels depend on the potential energy function, which dictates the way particles interact. In our scenario with quarks, the energy level \(E_n\) for a specific \(n\) is given by:\[E_{n} = \frac{1}{2m}\left(\frac{3mAh}{4}\right)^{2/3}n^{2/3}, \quad \text{where } n = 1, 2, 3, \ldots\]This formula shows that energy levels increase with \(n\), but not at a uniform rate. Specifically, as \(n\) becomes larger, the difference between consecutive energy levels decreases. This behavior contrasts with simpler systems like the harmonic oscillator, where energy levels are equally spaced. In a box potential, differences remain constant due to the linear nature, while in our quark system, this characteristic decreases due to the cubic root dependence.
Understanding these levels helps researchers predict particle properties, behavior, and interactions in particle physics experiments.

potential energy function

The potential energy function \(U(x) = A|x|\) represents how the quark and antiquark interact as a function of distance \(x\). This form, known as a linear potential, is an idealized way to understand the forces between quarks. In particle physics, potential energy functions describe forces that influence particle behavior, including attraction and repulsion.
This specific potential is 'linear,' meaning it's directly proportional to the distance between the particles. The constant \(A\) modifies the strength of interactions and is a crucial factor in calculating energy levels.
To determine the turning points, where a particle stops and changes direction, we set the energy \(E\) equal to \(U(x)\). According to classical mechanics, these are the points where kinetic energy reaches zero. For our linear potential, the turning points can be found where:\[x = \pm \frac{E}{A}\]These turning points help define the boundaries for the WKB approximation when calculating energy levels.
In practical terms, understanding the potential energy function allows us to predict how quarks within particles like \(\Psi\) behave, providing insights into the underlying physics of these fundamental particles.