A scientist has devised a new method of isolating individual particles. He claims that his method enables him to detect simultaneously the position of a particle along an axis with a standard deviation of 0.12 nm and its momentum component along the axis with a standard deviation of 3.0 x 10^-25 kgm/s. Evaluate the validity of this claim.
To evaluate the validity of the scientist's claim, we can use the Heisenberg uncertainty principle, which states that there is a fundamental limit to the precision with which certain pairs of physical properties, such as position and momentum, can be known simultaneously.
According to the Heisenberg uncertainty principle, for any given pair of complementary properties, the product of their standard deviations must be greater than or equal to a certain value, known as the reduced Planck's constant, denoted as h-bar (ħ).
The uncertainty principle is mathematically represented as:
Δx * Δp ≥ ħ/2
Where Δx represents the uncertainty in position and Δp represents the uncertainty in momentum.
In this case, the standard deviation of position (Δx) is given as 0.12 nm, and the standard deviation of momentum (Δp) is given as 3.0 x 10^-25 kgm/s.
Let's plug these values into the uncertainty principle equation:
(0.12 nm) * (3.0 x 10^-25 kgm/s) ≥ ħ/2
Now, we need to convert the units to a consistent system. We can convert nanometers (nm) to meters (m) and keep the momentum in kgm/s:
(0.12 x 10^-9 m) * (3.0 x 10^-25 kgm/s) ≥ ħ/2
Simplifying:
3.6 x 10^-34 mkg/s ≥ ħ/2
Comparing both sides of the equation, we find that the left-hand side (LHS) is much smaller than the right-hand side (RHS):
LHS << RHS
This indicates that the uncertainty principle is not violated. In other words, the product of the standard deviations of position and momentum is greater than or equal to the reduced Planck's constant, as required by the uncertainty principle.
Hence, the scientist's claim that they can simultaneously detect the position and momentum of particles with the given standard deviations is valid, considering the constraints imposed by the Heisenberg uncertainty principle.