What is Negative Probability and its Physical Interpretation?

[somy]

I have noticed a formula in which Cn (the probability density of the nth state ) was somthing like this:

Cn=1/ih*(...)

The probability of this state is then negative.
Can someone tell me about the physical interpretation of negative probability?
Thanks a lot.

 

[Norman]

I have noticed a formula in which Cn (the probability density of the nth state ) was somthing like this:

Cn=1/ih*(...)

The probability of this state is then negative.
Can someone tell me about the physical interpretation of negative probability?
Thanks a lot.

I am not sure but $C_n = \frac{1}{i \hbar}$ is a complex number. Typically to get the probability density for a state you take the complex conjugate wavefunction times the wavefunction:
$$\newcommand{\mean}[1]{{<\!\!{#1}\!\!>}}<br /> \newcommand{\braket}[2]{{<\!\!{#1|#2}\!\!>}}<br /> \newcommand{\braketop}[3]{{<\!\!{#1|\hat{#2}|#3}\!\!>}}<br /> \braket{\Psi}{\Psi} \equiv \int \Psi^*(x) \Psi(x)\,dx$$

 

[quantumdude]

That's right, Norman. C_n is the amplitude of that eigenstate, and it is complex. The probability of finding the particle in the n^th state is |C_n|², provided that the eigenfunctions in the overall wavefunction are all orthonormal.

 

[jcsd]

Though that doesn't mean to say that for some wave equations the probability density associated with them is always necessarily positive...

 

[quantumdude]

Though that doesn't mean to say that for some wave equations the probability density associated with them is always necessarily positive...

Yes, we'd need to know if he is talking about a Schrödinger wavefunction or a Klein-Gordon wavefunction.

In the latter case, the probability density is not positive definite, and it was at first thought that the KG equation was fatally flawed for that reason. But later KG was brought back to life by an re-interpretation (I forget by whom) of the KG probability density as an electric charge density (which of course is not required to be positive definite).

 

[yxgao]

I have seen negative probabilities as well - what does it mean physically?

 

[somy]

Thanks guys!
I did a silly mistake!
By the way Tom, can you tell me more about KG equations?
Thanks a lot.
Somy

 

[quantumdude]

1. The Klein-Gordon Equation
The Klein-Gordon (KG) equation was the first attempt at formulating relativistic quantum mechanics. Start from the relativistic energy-momentum relation for a free particle (in natural units):

p²+m²=E².

Now take the usual quantization rules:

p=-igrad, E=i(∂/∂t).

Plugging those into the energy-momentum relation gives the KG equation:

[-(grad)²+m²]φ(x,t)=(-i∂/∂t)φ(x,t)

Now if we define the 4-gradient and 4-position as follows:

∂_μ=(-grad,∂/∂t)
x_μ=(x,-t),

we can write the KG equation in manifestly covariant form:

[∂_μ∂^μ+m²]φ(x_μ)=0

2. The KG Probability 4-Current

Start with the system of the KG equation and its complex conjugate:

[∂_μ∂^μ+m²]φ(x_μ)=0
[∂_μ∂^μ+m²]φ^*(x_μ)=0

Now left-multiply the first equation by φ^*(x_μ) and the second equation by φ(x_μ), and subtract them. Suppressing the functional dependence of φ on x_μ, we have:

φ^*∂_μ∂^μφ-φ∂_μ∂^μφ^*=0,

which can be rewritten as:

∂_μ(φ^*∂^μφ-φ∂^μφ^*[/color])=0.

This is the KG continuity equation, and the quantity in blue[/color] is the KG 4-current density, whose timelike piece is the KG probability density. As you can see, it is not positive definite.

 

[somy]

Thank Tom.
It was very clear and useful.