A.) what is the minimum uncertainty in an electron's velocity (δmin) if the position is known within 13 å? b.) what is the minimum uncertainty in a helium atom's velocity (δmin) if the position is known within 1.0 å?

Answer :

(a) The minimum uncertainty in an electron's velocity is,

(b) The minimum uncertainty in a helium atom's velocity is,

Explanation :

According to the Heisenberg's uncertainty principle,

...........(1)

where,

= uncertainty in position

= uncertainty in momentum

h = Planck's constant

And as we know that the momentum is the product of mass and velocity of an object.

or,

     .......(2)

Equating 1 and 2, we get:

(a) Given:

m = mass of electron =

h = Planck's constant =

=

conversion used :

Now put all the given values in the above formula, we get:

The minimum uncertainty in an electron's velocity is,

(b) Given:

m = mass of helium atom =

h = Planck's constant =

=

conversion used :

Now put all the given values in the above formula, we get:

The minimum uncertainty in a helium atom's velocity is,