Klein-Gordon equation and continuity equation

In summary, the Klein-Gordon equation is a fundamental relativistic wave equation that describes scalar fields, such as spin-0 particles, in quantum field theory. It is derived from the principles of quantum mechanics and special relativity, incorporating a mass term and leading to solutions that represent particles and antiparticles. The continuity equation, on the other hand, is a mathematical statement that expresses the conservation of probability in quantum mechanics, indicating that the probability density and current are conserved over time. Together, these concepts are essential in understanding the behavior and interactions of particles in the framework of quantum field theory.
  • #1
dyn
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Hi
I am using the textbook "Modern Particle Physics" by Thomson. Working from the K-G equation and comparing with the continuity equation he states that the probability density is given by

ρ = i ( ψ*(∂ψ/∂t) - ψ(∂ψ*/∂t) )

He then states that the factor of i is included to ensure that the probability density is real. My question is - why does the factor of i make this real. This implies the quantity inside the bracket is pure imaginary. Why is that true ? ψ could be real , complex or pure imaginary. It is just a general wavefunction
Thanks
 
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  • #2
dyn said:
This implies the quantity inside the bracket is pure imaginary.
Yes. You can verify this easily by writing ##\psi = \psi_R + i \psi_I##, where ##\psi_R## and ##\psi_I## are real, and computing the quantity inside the bracket. You should find that all of the real parts cancel.

dyn said:
ψ could be real , complex or pure imaginary.
Yes, but the combination of terms inside the bracket is not ##\psi## and, as above, can be shown to be pure imaginary for a general ##\psi##.
 
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  • #3
Thank you very much for your help
 
  • #4
A real KG field describes an uncharged particle, and thus for this case ##\rho=0##.

In the case of charged fields, ##\rho## is the time-component of the Noether charge from invariance under ##\psi \rightarrow \exp(-\mathrm{i} \alpha) \psi, \quad \psi^* \rightarrow \exp(+\mathrm{i} \alpha) \psi^*##,
$$j_{\mu} = \mathrm{i} (\psi^* \partial_{\mu} \psi - \psi \partial_{\mu} \psi^*).$$
That's a real vector field, which is immediately clear when taking the complex conjugate of this definition.

That it's conserved follows from the (free) KG equation,
$$\partial_{\mu} j^{\mu} = \mathrm{i} (\psi^* \Box \psi-\psi \Box \psi^*) = -\mathrm{i} m^2 (\psi^* \psi-\psi \psi^*)=0.$$
 
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  • #5
dyn said:
ψ could be real
Could it be purely real? I don't think Schrödinger equation has purely real solutions.
 
  • #6
Sure, the KG equation is
$$(\Box+m^2) \psi=0$$
and thus has real solutions.

The time-dependent Schrödinger equation has a complex coefficient and thus has no real solutions.
 
  • #7
vanhees71 said:
Sure, the KG equation is
$$(\Box+m^2) \psi=0$$
and thus has real solutions.

The time-dependent Schrödinger equation has a complex coefficient and thus has no real solutions.
Ah, I see. So, when ##\psi## is real, then ##j_{\mu} = \mathrm{i} (\psi^* \partial_{\mu} \psi - \psi \partial_{\mu} \psi^*) = 0## and, in the OP, ##\rho=0##.
 
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FAQ: Klein-Gordon equation and continuity equation

What is the Klein-Gordon equation?

The Klein-Gordon equation is a relativistic wave equation for spin-0 particles. It is given by the equation (∂²/∂t² - ∇² + m²)φ = 0, where φ is the scalar field, m is the mass of the particle, ∂²/∂t² is the second time derivative, and ∇² is the Laplacian operator. It generalizes the Schrödinger equation to be consistent with special relativity.

How does the Klein-Gordon equation relate to quantum mechanics?

The Klein-Gordon equation extends quantum mechanics to include relativistic effects. While the Schrödinger equation is only applicable for non-relativistic particles, the Klein-Gordon equation incorporates the principles of special relativity, making it suitable for describing particles moving at speeds close to the speed of light.

What is the continuity equation in the context of the Klein-Gordon equation?

The continuity equation in the context of the Klein-Gordon equation ensures the conservation of probability. It is derived from the Klein-Gordon equation and is given by ∂ρ/∂t + ∇·j = 0, where ρ is the probability density and j is the probability current. This equation implies that the total probability is conserved over time.

How is the probability density defined for the Klein-Gordon equation?

The probability density ρ for the Klein-Gordon equation is defined as ρ = i(φ*∂φ/∂t - φ∂φ*/∂t), where φ is the scalar field and φ* is its complex conjugate. This definition ensures that ρ is real and can be interpreted as a probability density.

What are the differences between the Klein-Gordon equation and the Schrödinger equation?

The primary difference between the Klein-Gordon equation and the Schrödinger equation is that the former is relativistic while the latter is non-relativistic. The Klein-Gordon equation includes second-order time derivatives and is suitable for describing particles with relativistic speeds, while the Schrödinger equation includes first-order time derivatives and is applicable only to particles moving at much slower speeds. Additionally, the Klein-Gordon equation can describe particles with zero or integer spin, whereas the Schrödinger equation is typically used for spin-1/2 particles.

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