UNCERTAINTY IN POSITION - VELOCITY - MOMENTUM
IIT JEE - NEET

Question: Uncertainty in the position of an electron (mass = 9.1 x 10^{-34} kg) moving with a velocity 300 m s^{-1} , accurate up to 0.001%, will be: (h= 6.63 x 10^{-34} J s)

(AIEEE 2006)

a) 19.2 x 10^{-2 }m

b) 5.76 x 10^{-2 }m

c) 1.92 x 10^{-2 }m

d) 3.84 x 10^{-2 }m

Answer with Logic:

According to Heisenberg's uncertainty principle, the product of uncertainties in position (Δx) and velocity (Δv) is always equal to or greater than h/nπ.

i.e.

What is the meaning of "accurate up to 0.001%"?

It indicates that the uncertainty in velocity is 0.001% of actual value, 300 m s^{-1}.

i.e.

uncertainty in velocity (Δv) = 300 x 0.001 / 100 = 3 x 10^{-3 } m s^{-1}

Solution:

The uncertainty in position (Δx) can be calculated as follows:

Note: n = 4 for electron.

Conclusion:

The correct option is "c".

Related questions

1) If the uncertainties in velocities of two particles A and B with mass 1.0 x 10^{-27} kg and 1.0 x 10^{-31} kg respectively are same, the ratio of uncertainties in the positions of A and B is:

(EAMCET 2011-E)

a) 1000 :1

b) 10,000:1

c) 1: 1000

d) 1:10,000

Logic & solution:

The uncertainty in position, Δx is given by the expression:

The ratio of uncertainties in positions for A and B is:

Note: h/nπ is constant and Δv values are same.

Conclusion:

Correct option is "c".

2) The uncertainties in the velocities of two particles A and B are 0.05 and 0.02 m s^{-1} respectively. The mass of B is five times to that of mass A. What is the ratio of uncertainties?

(EAMCET 2006-E)

a) 2

b) 0.25

c) 4

d) 1

Logic & solution:

The uncertainty in position, Δx is given by the expression:

The ratio of uncertainties in positions for A and B is:

Note: m_{B} = 5m_{A}

Conclusion:

Correct option is "a".

3) Uncertainty in position of a particle of 25 g in space is 10^{-5} m. Hence, uncertainty in velocity (m s^{-1}) is: ( Planck’s constant, h = 6.6 x 10^{-34} J s)

(AIEEE 2002)

a) 2.1 x 10^{-28}

b) 2.1 x 10^{-34}

c) 0.5 x 10^{-34}

d) 5.0 x 10^{-24}

Logic & solution:

The uncertainty in velocity, Δv is given by the expression:

Conclusion:

Correct option is "a".

4) In an atom, an electron is moving with a speed of 600 m/s with an accuracy of 0.005%. Certainty with which the position of the electron can be located is: (h=6.6 x 10^{-34} kg m^{2 }s^{-1}; mass of electron, e_{m} = 9.1 x 10^{-31}
kg )

(AIEEE 2009)

1) 1.52 x 10^{-4 }m

2) 5.10 x 10^{-3} m

3) 1.92 x 10^{-3 }m

4) 3.84 x 10^{-3 }m

Logic & solution:

Since the accuracy in velocity is 0.005%, uncertainty in velocity, Δv = 600 x 0.005 / 100 = 3 x 10^{-2} m s^{-1}.

Uncertainty in position will be:

Conclusion:

Correct option is "3".

Homework

1) State Heisenberg's uncertainty principle.

2) What will be the uncertainty in momentum of electron if the uncertainty in its position is zero?