The solutions of a free particle are :a)Momentum eigen functions with only negative eigenvaluesb)Energy and momentum eigen functions with both positive and negative momentum eigenvalues.c)Energy and momentum eigen functions with positive momentum eigenvaluesd)Only energy eigen functions.Correct answer is option 'B'. Can you explain this answer?

Question

Answer

Introduction:
The solutions of a free particle refer to the possible wavefunctions that describe the motion of a particle that is not subject to any external forces. These solutions are obtained by solving the time-independent Schrödinger equation for a particle with no potential energy.

Explanation:
The time-independent Schrödinger equation for a free particle is given by:

Ĥψ = Eψ

Where:
- Ĥ represents the Hamiltonian operator, which describes the total energy of the particle.
- ψ represents the wavefunction of the particle.
- E represents the energy eigenvalue associated with the wavefunction.

Energy eigenfunctions:
The solutions to the Schrödinger equation are known as energy eigenfunctions. These are wavefunctions that satisfy the equation and correspond to specific energy eigenvalues. In the case of a free particle, the energy eigenvalues are continuous and can take on any positive value. Therefore, the solutions include energy eigenfunctions with positive energy eigenvalues.

Momentum eigenfunctions:
In quantum mechanics, momentum is represented by the momentum operator, p. The momentum eigenfunctions are the solutions to the momentum operator acting on the wavefunction. For a free particle, the momentum eigenvalues can be positive or negative, as the particle can move in either direction. Therefore, the solutions include momentum eigenfunctions with both positive and negative momentum eigenvalues.

Energy and momentum eigenfunctions:
Since the solutions to the Schrödinger equation for a free particle include both energy and momentum eigenfunctions, option B is correct. These wavefunctions describe the particle's motion in terms of both its energy and momentum.

Momentum eigenfunctions with only negative eigenvalues:
Option A is incorrect because the solutions to the Schrödinger equation for a free particle include momentum eigenfunctions with both positive and negative eigenvalues.

Only energy eigenfunctions:
Option D is incorrect because the solutions to the Schrödinger equation for a free particle include both energy and momentum eigenfunctions.

Conclusion:
In summary, the solutions of a free particle include energy and momentum eigenfunctions with both positive and negative momentum eigenvalues. These wavefunctions describe the particle's motion in terms of both its energy and momentum and satisfy the time-independent Schrödinger equation.

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