Quantum Mechanics

To solve the problem of determining the change in the de Broglie wavelength of an electron in an electric field, we start by analyzing its motion under the field's influence.

Initial Conditions:

Initial velocity of the electron:

Initial de Broglie wavelength: where is Planck’s constant, is the mass of the electron, and is the initial velocity.

Effect of Electric Field:

The electron enters a uniform electric field:

Force on the electron: Acceleration:

Velocity After Time :

New de Broglie Wavelength:

Simplified Expression:

Factor and simplify:

Conclusion:

The de Broglie wavelength of the electron at time in the electric field is:

Step 1: Known Information.
Uncertainty in velocity of particle :
Uncertainty in velocity of particle :
Mass of particle is four times the mass of particle : We need to find the ratio of uncertainties in their positions ( and ).
Step 2: Heisenberg Uncertainty Principle.
The Heisenberg Uncertainty Principle states: where:
is the uncertainty in position,
is the uncertainty in momentum,
is the reduced Planck's constant.
Momentum is given by: Thus, the uncertainty in momentum is: Step 3: Apply the Uncertainty Principle for Each Particle.
For particle :
Substitute : Rearrange to solve for : For particle : Substitute : Rearrange to solve for : Step 4: Ratio of Uncertainties in Position.
The ratio of uncertainties in position is: Simplify: Step 5: Substitute Given Values.



Substitute these values into the ratio: Simplify: Final Answer: