This problem is related to the quantum mechanical model for a one-dimensional \"

Question

This problem is related to the quantum mechanical model for a one-dimensional "particle in a box", which assumes a potential energy function that is zero for coordinates x between 0 and L, the length of the box, and infinite everywhere else.

a) Describe the behavior you expect for a classical particle with some fixed kinetic energy that is palced in the same potential energy function as the quantum "particle in a box", i.e. a well with infinite potential energy walls.

b) In class, we demonstrated that quantum mechanical expectation value for the momentum p was zero for all eigenfunctions of the particle-in-a-box, even though the kinetic energy, which includes the expectation value for the squared momentum p2 is nonzero for these functions. How do you reconcile these two observations? (your answer to (a) may help).

Explanation / Answer

a) a clasical particl within the infinet well can have any amount of KE energy and is confined to the well. Its momentum is given by (2mE)1/2 . It s motion the postion and monetum at any given instant of oint canbe correctly predicted.

b) Enrgy of the quantum particle is a sharp quantity and can have qunatized values and canbe measure precisely.

The postion and momentum of the quantum particle are fuzzy values an are governed by uncerntianity principle. We can only state the probabilty of these quanties having certian value and not its precies value. The expctaion value is nothing but the average value of it summed over all the probabilities.

The average or the expectation value of the position <x> in the box is L/2 , this is beacuse the probabilit of finding the particle wither side from the middle is equal.

Though the particle has a non-zero kinetic energy it average position is confined to the middle of the box henc ete average momentum is 0. This makes sense in the classical way too. Further momentum is a vector quantity and the momentum on the right side cancel up with on the left side as they have equal probability but opposite in direction.

When we compute <p2> the expcetation values or the average value of the squared quantity it is non-zero because all the quantities either on left or right become +ve and add up to a non-zero value.

Related Questions

Navigate

• Browse (All)
• Browse T
• Subjects

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.