Quantum Mechanics – PHY202
Tutorial Questions 2: Answers
Sketching Wavefunctions
1.
Parity: a wavefunction has definite parity if ψ(−x) = ±ψ(x); this requires a symmetric
potential, V(−x) = V(x).
2.
Minimum energy implies maximum wavelength, so no nodes. No nodes condition
implies parity is not odd (odd parity requires a central node), so if the wavefunction has
definite parity it must be even.
3.
KE = p2/2m ∝ 1/λ2. As V increases, KE (= E – V) decreases, so λ increases.
4.
“Sketches” are stills from the 1D quantum applet at www.falstad.com/mathphysics.html
Answers to questions:
(b) Dip in centre will become a node, i.e.
ψ(0) → 0 as V0 → ∞.
(c) Fewer and fewer bound states, until the
well eventually disappears altogether
leaving
a travelling wave solution.
5(a) Looks like 4(b), but with infinite sides –
wavefunction goes sharply to zero instead of
tailing off, so outside potential must be
infinite.
(b) Harmonic oscillator – which we will cover
later in the course.
(c) Well with a step in it: since wavelength on
right is twice left wavelength, KE on right
must be ¼ of KE on left, and since sketch
seems to show same number of peaks in each,
deeper well is ~1/3 width of shallower well.
Sides of well are infinite, since wavefunction
goes sharply to zero.
Operators and Measurement
6.
Give examples of eigenvalue equations, stating eigenvalues and eigenfunctions, for
(a)
the energy operator
(b)
the momentum operator
(c)
the parity operator
Obvious examples for (a) are the infinite square well 0 < x < L,
(since V(x) = 0),
where
and
, or the
infinite square well −L/2 < x < L/2 (results see Tutorial Questions 1, Q3). They have
also done the finite square well and the harmonic oscillator.
Obvious example for the momentum operator,
where
, is
the plane wave u(x) = Aeikx, with eigenvalue p = k.
For parity,
, the eigenvalues are P = ±1, and any symmetric or
antisymmetric solution is an eigenfunction. The obvious concrete example is the
symmetric infinite square well.
7.
Suppose that φ (x) is a travelling wave, Aexp(ikx).
1
(a)
Show that this is an eigenstate of the momentum operator. What is its momentum?
See above.
(b)
What is
φ (x) = Ae−ikx.
? (where
is the parity operator).
2
(c)
Using φ (x) and φ (x) construct states of definite parity.
1
2
ψ(x) = (φ1 ± φ2)/√2, the + sign having P = +1 and vice versa.
(d) Show that these states of definite parity satisfy the parity eigenvalue equation.
Trivial.
8.
A particle in an infinite square well, V(x) = 0 for 0 < x < L, V(x) = ∞ otherwise, has
the time independent wavefunction
(a)
By exploiting the orthonormality of the expansion functions, find the value of the
normalization factor A.
Write the equation as
eigenstates properly. Then we have
to normalise the
, from which
.
(b)
If a measurement of the energy is made, what are the possible results? What is the
probability associated with each result?
and
, with probabilities 0.8 and 0.2 respectively.
(c)
Using the results of (b) deduce the average energy and express it as a multiple of
the energy E1 of the lowest eigenstate.
0.8E1 + 0.2(4E1) = 1.6E1.
(d)
Calculate the expectation value of E using
and verify that this is identical to the result of (c).
, so
.
Multiplying the integrand out and using double angle formulae reduces the
integral to
;
all the cos terms contribute zero (they integrate to sines, which are zero at 0 and L
by the boundary conditions), so we wind up with
,
as expected.
(e)
If the energy is measured, as in (b), what is the form of the wavefunction after the
measurement?
Either
(80% probability) or
(20% probability), depending
on the outcome of the measurement.
(f)
If the energy is immediately re-measured, what will be the probabilities of the
possible outcomes now?
The re-measurement gives the same value as the original measurement, with
100% probability.
9.
What does it mean to say that certain operators commute? Give examples of operators
that commute and of operators that do not commute. (Hint, see also Q.2.)
Definition: operators commute if
, i.e. the order does not matter.
Significance: observables corresponding to commuting operators can be measured
simultaneously, whereas those corresponding to non-commuting operators have an
uncertainty principle.
Examples: commuting pairs, momentum and energy, energy and parity; non-commuting
pairs, momentum and position, energy and time.
10.
Use the uncertainty principle to explain, in the simplest terms, why a zero point energy
is observed for any particle in a bound state.
A bound state is by definition confined to a certain region of space, so Δx ≠ ∞.
Therefore, since Δx Δp ~ , we have a minimum Δp ~ /L where L is the width of the
bound state. But Δp = 〈p2〉 − 〈p〉2, so the minimum value of 〈p2〉 is ~ 2/L2. Therefore
we must have a minimum energy ~ 2/2mL2, from E = p2/2m.