Today: Eigenstates vs.Superposition states
Measurements
Written HW11 due on Mon@ noon.
Three pillars of QM
1.  Wavefunction
2.  Schrodinger equation
3.  Observables: Things we, humans, can
observe and/or measure: Position,
momentum, velocity, force, color, rigidity,
angular momentum, etc.
Observables
Position, momentum, velocity, force, color,
rigidity, angular momentum, etc.
Values of observables are numbers with units
How to extract them from the wavefunction???
If we know the wavefunction,
how to predict outcomes of
experiments?
Example 1: The experiment measures the
energy of an electron in an infinite square
well.
V(x)
Brief review: energy eigenstates
in the infinite square well:
Ψ3(x,t)
Ψ2(x,t)
A state with a single energy En is
called an “energy eigenstate.”
We found that |Ψn(x,t)|2 does not
depend on time:
=1
Ψ1(x,t)
(no time dependence)
If the wavefunction of an electron is an
energy eigenstate,
then its energy is given by the energy
eigenvalue, En
What if
Ψ(x,t) =
1
2
Ψn1 (x,t) +
1
2
Ψn 2 (x,t) ?
a)  You will measure either En1 or En2 with equal probability
b)  You will measure its energy to be (En1+En2)/2
c)  The experiment will not give an answer
€
Superposition States
An electron is in the state
,where
Ψ1(x,t) = ψ1(x)e–iE1t/ and Ψ2(x,t) = ψ2(x)e–iE2t/
are the ground state and first excited state of
the infinite square well, and
Is the probability density |Ψ(x,t)|2 time
dependent?
a. Yes b. No
c. Depends (on what?)
Probability density doesn’t change in time for energy
eigenstates, but it does for superpositions of
energy eigenstates!
Probability density:
Cross terms oscillate between
constructive and destructive interference!
Quantum Bound States sim
In simple terms: What does it mean for a
particle to be in a superposition of states
Ψ1(x,t) and Ψ2(x,t)?
A.  There are two particles, one described by Ψ1(x,t) and
the other described by Ψ2(x,t), that travel together in a
packet.
B.  The probability density of finding the one particle at
position x at time t is given by the absolute square of
the sum of the two wave functions, each multiplied by
some factor.
C.  The one particle is located at a position somewhere in
between the position described by Ψ1(x,t) and the
position described by Ψ2(x,t).
D.  The one particle has an energy somewhere in
between the energies E1 and E2.
E.  More than one of the answers above is true.
B: The probability density of finding the particle at position x
at time t is given by the absolute square of the sum of the
two wave functions, each multiplied by some factor.
i.e. something of the form |Ψ|2 = |c1Ψ1 + c2Ψ2 |2. The
constants are chosen so that
(“one particle”)
Easiest example I can think of: one electron at the double-slit
ψ1
ψ2
Probability density of detecting it at the screen:
|Ψ|2 = |c1Ψ1 + c2Ψ2 |2
0
Note: Here, ψ1 and ψ2 have same energy!
à (E2-E1)=0. à interference pattern has no time dependence
Superposition state:
Mathematically easy!
with:
(i.e. only one particle!)
Note: Ψ(x,t) is a linear combination of solutions
(and hence itself a solution) to the Schrödinger eqn.
But what does this mean??!
à Need to talk about ‘measurements.’
A measurement will tell us where the
particle is or what energy it has.
Still only one particle?
=1/2
=1/2
even
odd
=0
Still only one particle!
Superposition state
x
Superposition state
∞
∞
Ψ2(x,t)
Ψ1(x,t)
x
Superposition state
|Ψn(x,t)|2 = |ψn(x)|2 : time independent
But:
Ψ2(x,t)
Ψ1(x,t)
x