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koustav
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- Second order time derivative
What problem actually arises when we take the second order time derivative in KG equation
The textbook simply mentions that the probability density is not positive definate because of the second order time derivative.so how actually the second derivative creates a problem for probabilistic interpretationvanhees71 said:We'd need a bit more detail, to which problems you are referring to. Sometimes it's claimed that the "problem" with "negative energy solutions" of the time-independent Klein-Gordon equations arises because of the 2nd-order time derivative, and then old-fashioned books make an argument that for this reason one wants the 1st-order Dirac equation. The modern answer to all these problems is, of course, quantum field theory and the reinterpretation of negative-frequency modes as antiparticle states with positive energy (just writing a creation instead of an annihilation operator in the mode-decomposition of the field operator, aka Feynman-Stückelberg trick).
Which textbook?koustav said:The textbook simply mentions that the probability density is not positive definate because of the second order time derivative.so how actually the second derivative creates a problem for probabilistic interpretation
How are the probabilities computted? If it is ##\int |\psi|^2 dx## again, then it is always non-negative.Gaussian97 said:The problem is to try to use KG equation as an immediate relativistic substitute for the Schrödinger equation. In classical QM you know that the wave function that describes a particle must be a solution of the equation
$$\left(i\hbar \frac{\partial}{\partial t} + \frac{\hbar^2}{2m}\frac{\partial^2}{\partial x^2}\right)\psi=0$$
and then, the probability of the particle to be in some region is given by
$$\int |\psi|^2 dx$$
If you want to describe a relativistic particle just by using KG equation, i.e. that now the wave function must be a solution of
$$\left(\hbar^2\frac{\partial^2}{\partial t^2} - c^2\hbar^2\frac{\partial^2}{\partial x^2} + m^2 c^4\right)\psi=0$$
then you have problems because the probabilities become negative.
So constructing a relativistic QM is not as easy as that, you need to change some fundaments. And that's why one needs Quantum Field Theory.
martinbn said:How are the probabilities computted? If it is ##\int |\psi|^2 dx## again, then it is always non-negative.
But then it's not conserved in time, which is inconsistent because the sum of all probabilities ##\int_{-\infty}^{\infty} |\psi|^2 dx## should be equal to 1, at any time.martinbn said:How are the probabilities computted? If it is ##\int |\psi|^2 dx## again, then it is always non-negative.
Yes, my question was about that part, where he wrote that he probabilities are negative.Demystifier said:But then it's not conserved in time, which is inconsistent because the sum of all probabilities ##\int_{-\infty}^{\infty} |\psi|^2 dx## should be equal to 1, at any time.
martinbn said:Yes, my question was about that part, where he wrote that he probabilities are negative.
Oh yes, completely true, my fault. Of course, the definition of probability density must change. Anyway, I think @stevendaryl has already answered your question.martinbn said:How are the probabilities computted? If it is ##\int |\psi|^2 dx## again, then it is always non-negative.
The invariance of the Lagrangian of the Klein-Gordon equation for the complex scalar field leads, via Noether's, theorem to the conserved currentstevendaryl said:Another way to put my point about the Klein Gordon equation is this:
You have two choices, and neither works: If you interpret ##\rho = \psi^* \psi## as a probability density, then that doesn't work because the total probability isn't conserved for Klein-Gordon. If you interpret ##\rho = \psi^* \frac{d\psi}{dt} - \frac{d\psi^*}{dt}## as a probability density, then it is in fact conserved, but it can be negative.
I guess you can consistently interpret ##\rho = \psi^* \frac{d\psi}{dt} - \frac{d\psi^*}{dt}## as a charge density, with both positive and negative charges...
The Klein-Gordon equation is a relativistic wave equation that describes the behavior of spinless particles, such as the Higgs boson, in quantum field theory. It is a second-order partial differential equation that combines elements of both the Schrödinger equation and the relativistic energy-momentum relation.
The 2nd order time derivative in the Klein-Gordon equation represents the time evolution of the wave function of a particle. Solving for this derivative allows us to understand how the particle's wave function changes over time, and therefore, how the particle itself behaves and moves through space.
The Klein-Gordon equation is solved using various mathematical techniques, such as separation of variables, Fourier transforms, and Green's functions. These techniques allow us to find the wave function of a particle at any given time and location.
The Klein-Gordon equation has been used in various fields of physics, including particle physics, quantum mechanics, and cosmology. It has also been applied in other areas such as quantum field theory, condensed matter physics, and nuclear physics. Additionally, the equation has been used to study the behavior of particles in accelerators and to predict the properties of new particles.
One limitation of the Klein-Gordon equation is that it only describes spinless particles. It also does not take into account the effects of external forces or interactions between particles. Additionally, the equation does not account for relativistic effects at very high energies, where a more complex equation, such as the Dirac equation, is needed.