# Relativistic particle moving in a potential

• PhysicsRock
PhysicsRock
Homework Statement
Consider a relativistic particle of mass ##m_0## in ##(1+1)##-spacetime dimensions. In an inertial frame of reference with spacetime coordinates ##(t,x)##, we define the potential as ##V(x) = \alpha x## with ##\alpha > 0##. The particle is at rest at time ##t=0## and it's position is ##x(0) = 0##. Determine the trajectory ##x(t)## of the particle.
Hint: Use the conservation of total energy.
Relevant Equations
##E_\text{tot} = \sqrt{ c^2 p^2 + m_0^2 c^4 } + \alpha x##
Since energy is conserved and the particle is initially at rest, we can determine that ##E(0) = m_0 c^2##, so

$$m_0 c^2 = \sqrt{ c^2 p^2 + m_0^2 c^4 } + \alpha x.$$

Squaring this eqation gives

$$m_0^2 c^4 = \alpha^2 x^2 + c^2 p^2 + m_0^2 c^4 + 2 \alpha x \sqrt{ c^2 p^2 + m_0^2 c^4 } \Rightarrow 0 = \alpha^2 x^2 + c^2 p^2 + 2 \alpha x ( E - \alpha x ).$$

Using ##p = \gamma m_0 \dot{x}##, I was able to simplify this equation to

$$0 = -\alpha^2 x^2 + \frac{E^2 \dot{x}^2}{c^2 - \dot{x}^2} + 2 \alpha E x$$

This is the point where I'm stuck. I have doubled checked and I'm pretty sure that this final expression is correct, however, I cannot guarantee that it actually is. If it is, I have no clue how to solve this equation.

PhysicsRock said:
$$0 = -\alpha^2 x^2 + \frac{E^2 \dot{x}^2}{c^2 - \dot{x}^2} + 2 \alpha E x$$
This expression is correct, and can be solved via integration using the following steps:
1. Solve for ##\dot{x}## to get the form ##\dot{x}\equiv\frac{dx}{dt}=f\left(x\right)##.
2. Rewrite this as ##\frac{dx}{f\left(x\right)}=dt##.
3. Integrate both sides.

vanhees71, PhysicsRock, topsquark and 1 other person
renormalize said:
This expression is correct, and can be solved via integration using the following steps:
1. Solve for ##\dot{x}## to get the form ##\dot{x}\equiv\frac{dx}{dt}=f\left(x\right)##.
2. Rewrite this as ##\frac{dx}{f\left(x\right)}=dt##.
3. Integrate both sides.
Thank you for the hint. I'll try to solve it.

## What is a relativistic particle?

A relativistic particle is one that moves at a significant fraction of the speed of light, such that its behavior must be described by the principles of Einstein's theory of relativity rather than classical mechanics. This includes effects such as time dilation, length contraction, and the increase of mass with velocity.

## How does a potential affect a relativistic particle?

A potential affects a relativistic particle by altering its energy and momentum. The interaction between the particle and the potential must be described using the relativistic form of the equations of motion, which take into account the relativistic energy-momentum relationship. This can lead to phenomena like the Klein paradox or tunneling effects that differ from non-relativistic cases.

## What is the Klein-Gordon equation?

The Klein-Gordon equation is a relativistic wave equation for spin-0 particles. It extends the Schrödinger equation to be consistent with special relativity. The equation is given by (∂²/∂t² - ∇² + m²)ψ = 0, where ψ is the wave function of the particle, m is its mass, and natural units (ℏ = c = 1) are often used.

## How does the Dirac equation describe a relativistic particle in a potential?

The Dirac equation describes relativistic spin-1/2 particles, such as electrons, in a potential. It incorporates both the principles of quantum mechanics and special relativity. The equation is given by (iγ^μ∂_μ - m)ψ = 0, where γ^μ are the gamma matrices, ∂_μ represents the spacetime derivatives, m is the mass, and ψ is the four-component spinor wave function. When a potential is present, it is included in the Hamiltonian, modifying the equation accordingly.

## What are some key differences between relativistic and non-relativistic quantum mechanics?

Key differences between relativistic and non-relativistic quantum mechanics include the treatment of time and space, the necessity to consider spin explicitly in the relativistic case, and the presence of antiparticles. Relativistic quantum mechanics also predicts phenomena such as pair production and annihilation, which have no counterpart in non-relativistic theory. Additionally, the equations of motion are derived from relativistic energy-momentum relations, leading to different behavior at high velocities.

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