tion.- The output shouldn’t depend on amplitude (because a dif-
ferent amplitude might just mean an incorrectly normalized
state). - The output should be well defined when we superpose any two
states.
These requirements are hard to reconcile with the idea that the
output of the observable is just a real number representing the result
of the measurement. We could decree that the input wavefunction
is just required to be have the standard normalization, but there’s
no obvious way to define a standardization of phase. And suppose
we have a particle in a one-dimensional box, with the two lowest
energies beingE( ) = 1 and E( ) = 4. Then what should
we define for the superpositionE( + )? We could define it
to be the average, 2.5, but that isn’t even a possible value of the
measurement; in reality, the result of the measurement would be
either 1 or 4, with equal probability.
For a clue as to a better way to proceed, note the structure of the
time-independent Schr ̈odinger equation for a free particle, omitting
all constant factors likem, 2, and~. It isn’t (d^2 /dx^2 )Ψ =E, it’s
(d^2 /dx^2 )Ψ =EΨ. This fixes all the problems. For example, if
we change the phase of the wavefunction by flipping its sign, the
equation still holds with the same value ofE. This equation is a
specific example of a more general type of equation that looks like
operator(input) = number×input.Another, simpler example is (d/dx)f = 3f, which is satisfied if
f = Ae^3 x, whereAis any constant. Such an equation says that
applying the operator to the input just gives back theinput itself,
multiplied by some constant. For this reason, this type of equation is
called aneigenvalue equation, because “eigen” is the German word
for “self.” We say that 3 is the eigenvalue of the eigenvalue equation
(d/dx)f= 3f. In the time-independent Schr ̈odinger equation, the
eigenvalue is the energy, and a solution Ψ is called a state of definite
energy (or “eigenstate”).
All observables in quantum mechanics are described by opera-
tors such as derivatives. The second derivative (with the appropriate
factor of−h^2 / 2 m) is the kinetic energy operator in quantum me-
chanics. Given an operatorOthat describes a certain observable,
a state Ψ with a definite valuecof that observable is one for which
O(Ψ) =cΨ. Although it’s common to use parentheses when no-
tating functions, as in cos(π) =−1, they are optional, and we can
write cosπ=−1, so we will often use notations likeOΨ instead of
O(Ψ), but keep in mind that this not multiplication, just as cosπ
doesn’t mean multiplying cos byπ.
Section 14.6 The underlying structure of quantum mechanics, part 2 975