The wave-function of a particle at state ‘n’ is given by,
ψ(x) = √ 3 cos( 2x /a ), |x| ≤ a/2
0, |x| > a/2
What is the average momentum of the particle at the state ‘n’?
(a) zero
(b) 6ℏ/a
(c) ℏ/a
(d) none of these
Answers
Answer:
The expectation value is the probabilistic expected value of the result (measurement) of an experiment. It is not the most probable value of a measurement; indeed the expectation value may even have zero probability of occurring. The expected value (or expectation, mathematical expectation, mean, or first moment) refers to the value of a variable one would "expect" to find if one could repeat the random variable process an infinite number of times and take the average of the values obtained. More formally, the expected value is a weighted average of all possible values.
Average Energy of a Particle in a Box
If we generalize this conclusion, such integrals give the average value for any physical quantity by using the operator corresponding to that physical observable in the integral in Equation 3.7.4 . In the equation below, the symbol ⟨H⟩ is used to denote the average value for the total energy.
⟨H⟩=∫−∞∞ψ∗(x)H^ψ(x)dx=∫−∞∞ψ∗(x)KE^ψ(x)dx+∫−∞∞ψ∗(x)V^ψ(x)dx=∫−∞∞ψ∗(x)(−ℏ22m)∂2∂x2ψ(x)dxaverage kinetic energy+∫−∞∞ψ∗(x)V^(x)ψ(x)dxaverage potential energy(3.7.5)(3.7.6)(3.7.7)
The Hamiltonian operator consists of a kinetic energy term and a potential energy term. The kinetic energy operator involves differentiation of the wavefunction to the right of it. This step must be completed before multiplying by the complex conjugate of the wavefunction. The potential energy, however, usually depends only on position and not momentum (i.e., it involves conservative forces). The potential energy operator therefore only involves the coordinates of a particle and does not involve differentiation. For this reason we do not need to use a caret over V in Equation 3.7.7 .
Equation 3.7.7 can be simplified
⟨H⟩=⟨KE⟩+⟨V⟩(3.7.8)
The potential energy integral then involves only products of functions, and the order of multiplication does not affect the result, e.g. 6×4 = 4×6 = 24. This property is called the commutative property. The average potential energy therefore can be written as
⟨V⟩=∫−∞∞V(x)ψ∗(x)ψ(x)dx(3.7.9)
This integral is telling us to take the probability that the particle is in the interval dx at x , which is ψ∗(x)ψ(x)dx , multiply this probability by the potential energy at x , and sum (i.e., integrate) over all possible values of x . This procedure is just the way to calculate the average potential energy ⟨V⟩ of the particle.
Exercise 3.7.2 : Particle in Box
Evaluate the two integrals in Equation 3.7.7 for the PIB wavefunction ψ(x)=2L−−√sin(kx) with the potential function V(x)=0 from 0 to the length of a box L with k=πL .
Solution
The average kinetic energy is
⟨KE⟩=∫0L(2L−−√)sin(kx)(−ℏ22m)∂2∂x2(2L−−√)sin(kx)dx=(2L)∫0Lsin(kx)(−ℏ22m)∂∂xcos(kx)(k)dx=(2L)∫0Lsin(kx)(−ℏ22m)sin(kx)(k)(−k)dx=(2L)(k2ℏ22m)∫0Lsin2(kx)dx
We can solve this intragral using the standard half-angle represention from an integral table. Or we can recognize that we already did this integral when we normalized the PIB wavefunction by rewriting this integral:
⟨KE⟩=(2L)(k2ℏ22m)∫0Lsin2(kx)dx=(k2ℏ22m)∫0L(2L)sin2(kx)dx=(k2ℏ22m)∫0Lψ∗(x)ψ(x)dx1=k2ℏ22m
Thus, the average value for the total energy of this particular system is
Explanation: