Since you’ve already studied relativity, you’ve had carefully in-
culcated in you the idea that space and time are to be treated sym-
metrically, as parts of a more general thing called spacetime. The
differing results of examples 7 and 9 are clearly not consistent with
relativity. This is to be expected because the Schr ̈odinger equa-
tion is nonrelativistic (cf. self-check G, p. 905), and the principles
laid out in this section are the principles ofnonrelativisticquantum
mechanics.
Parity example 10
In freshman calculus you will have encountered the notion of even
and odd functions. In quantum mechanics, we can have even
and odd wavefunctions, and they can be distinguished from one
another using the parity operatorP. IfΨ(x) is a wavefunction,
thenPΨis a new wavefunction, call itΨ′, such thatΨ′(x) =Ψ(−x).
In other words, the parity operator flips the wavefunction across
the origin. (In three dimensions, we negate all three coordinates.)
States of definite parity are represented by wavefunctions that are
even (eigenvalue +1) or odd (−1).
States of definite angular momentum example 11
In section 14.2.4, p. 962, we saw that the kinetic energy of a
quantum mechanical rotor is proportional not to`^2 but instead
to`(`+ 1). This was justified qualitatively in terms of the solutions
of the Schrodinger equation for a particle on a sphere, but in fact ̈
there is a deeper reason, which is that the eigenvalues of the
orbital angular momentum operator turn out to be`(`+ 1). The
parity of such a state is (−1)`, which can be seen in figure h on
p. 928.
If we have two observables, it may or may not be possible to
measure them both on the same state and get exact and meaningful
results. Position and momentumpandxare incompatible observ-
ables, as expressed by the Heisenberg uncertainty principle. No state
is simultaneously a state of definitepand of definitex. The mag-
nitude of an angular momentumLand its component along some
axisLzare compatible. It is common to have a state that is simul-
taneously a state of definiteLand of definiteLz. Another example
ofincompatibleobservables isLzandLx, as proved on p. 922.14.6.2 The inner product
We’ve defined the normalization of a wavefunction as the re-
quirement∫+∞
−∞Ψ
∗Ψ dx= 1, which means that the total probability
that the particle issomewhere equals 1. (Another way of writing
Ψ∗Ψ would be|Ψ|^2 .) This assumes that the wavefunction is writ-
ten as a function of the positionx. But it is also possible to have
a wavefunction that depends on some other variable, such as spin
or momentum, or on some combination of variables, e.g., both the
spinsand the positionxof an electron, Ψ(x,s). We can also use
a wavefunction to describe a correlation between multiple particles,980 Chapter 14 Additional Topics in Quantum Physics