hello in this video we're going to take
a look at quantum superposition
the central story of the video is going
to be that quantum mechanics is linear
what we mean by that is that if two wave
functions
psi one and psi two which are independently
solutions
to the time-dependent Schroedinger equation
then any linear combination of them
is also a solution
that is wave functions of a linear
superposition
in general as we saw in a previous video
we can
write the following
if we can write some arbitrary wave
function down
we can decompose it in terms of energy
eigenstates
psi_n and
any linear combination like this will
also be a solution to the schrodinger
equation so it's not just a sum of two
it's a sum of any number of wave
functions which individually solve
the time dependent Schroedinger equation
let's clear the board just move this up
to the top
and just to reiterate if psi_n is a
solution to the time-dependent
Schroedinger equation
then any arbitrary function which is
written as a superposition a sum
of different psi_n weighted by
coefficients
f_n which are in general complex this is
also
a solution these are energy eigenstates
that we've been dealing with so far if
we perform
a measurement on f(x,t)
a measurement of energy this is where
things get a little bit strange in
quantum mechanics
we will always find exactly one of
the energy eigenstates
so let's write that down. A measurement
of the energy of
f(x,t) will reveal one energy E_n
so even though we prepare the state in a
superposition of different energy
eigenstates when we make a measurement
of the energy we only find one of the
energies
and as a result of that the wave
function will change as a result of that
measurement
it'll no longer be f(x,t) it will
then be the energy eigenstate
corresponding to that eigenenergy
after measuring eigenenergy eigenvalue
e_n
the state is psi_n(x,t)
and that's with certainty
whether you want to say that the state
has changed depends on your
interpretation of quantum mechanics
in the standard interpretation we tend
to teach at university, the Copenhagen
interpretation,
this process is called
'wave function collapse'
but this is an interpretational question
in the many worlds theory for example
wavefunction collapse does not exist
there is a different process for
explaining how a state prepared
in f(x,t) can, when measured
according to its energy, change into
psi_n(x,t) in fact it doesn't change
it just appears to change to us so
the probability when performing an
energy measurement on f
to find energy E_n and for the state
subsequently to be
psi_n is given by the modulus square of
the coefficient
assuming this is correctly normalized
which all
physical states in quantum mechanics
are
the probability to find the result E_n
in an energy measurement of f(x,t)
|f_n|^2 before the
measurement
after the measurement it will be in
state psi_n
and it'll have energy E_n with
probability one so the state really has
changed
we'll look at some of the more
philosophical interpretations
surrounding this
in a separate video for now let's
take a look at what this means for the
time dependence of states
let's just clear the board so let's take
a look at the time dependence
in the absence of measurement the
time dependence for state is dictated
entirely by the time dependent
Schroedinger equation
so we have that the state psi evolves
unitarily
according to the time-dependent Schroedinger
equation in the absence of measurement
the word unitarily here we'll see in
more detail what this means later on in
the course
but for now all you need to know is that
a unitary evolution of the wave function
simply preserves the normalization if
you start with a normalized wave
function which you must for a physical
state
it remains normalized for all subsequent
times
and this is built into the schrodinger
equation
okay so we'd like to look at the time
dependence of states remember if we have
an energy eigenstate
the probability density of that
eigenstate is time independent
so if this is our energy eigenstate
labeled with subscript n
then the probability density is this
so rho_n is modulus square of psi n
this is a function of time but the
result is completely equal to
phi_n only a function of
position x
and that's because of this form of the
time evolution
however when we take a quantum
superposition of two different energy
eigenstates
the result the resulting probability
density need not be time independent
so consider this state alpha psi one
plus b psi two where
psi one and psi two are different energy
eigen states
the probability density is this
it's equal to modulus psi squared again
and expanding
the product we find this result
and if we stick in the forms of the
energy eigenstates again
we find the result
so it takes this form from this term
here
psi one is time dependent but when we
take the modulus square of it because
it's an energy eigenstate the result is
the same as phi one squared
modulus which is time independent same
with phi two
but the cross-terms are in general
time dependent the result is still
real
because it's a probability density so it
better be real and we know it must be
real because we're taking the modulus
square
of some number which is always real
but
it may in general be time dependent even
though the states from which it's a sum
have separate time
independent probability
densities
okay now let's take a look at the
normalization of this superposition so
let's clear the board again
so look at normalization we'll pick the
same state as before
and remember that all physical states
must be normalized
which means that the modulus square of
the wave function integrated across all
of space
must equal one we ensure that our energy
eigenstates are normalized
and so this places a condition on the
possible alphas and betas we can take in
our superposition
in this case the condition is this
which equals again just expanding the
product into these four terms
but we know that not only are our energy
eigenstates normalized they're also
orthogonal to one another
so that condition by definition means
that this term is zero
because this is psi one and psi two
are
orthogonal so by definition this is zero
same with this term
and we also have that these two states
are normalized
so putting this together we find the
following
that is one equals the modulus of alpha
squared plus the modulus of beta squared
so this gives us our normalization that
must appear on
this wave function as the following that
is we now have a properly normalized
wave function
psi for arbitrary complex alpha and beta
if we perform an energy measurement on
psi
it's not itself in energy eigenstates we
won't certainly get
any particular energy we can say for
certainty that we won't get
any result other than E_1 or E_2 because
the amplitude for any other energy
eigenstate is zero
it didn't need to be but it is in this
particular choice
and the probability that we'll find
energy E_1 is given by
the modulus of alpha squared divided by
the normalization which would be this
thing squared
so it'll be modulus alpha squared
divided by alpha modulus
squared plus modulus beta squared and
similarly the probability for finding
energy E_2
will be the modulus of beta squared
divided by the normalization
so this guarantees that the total
probability to find
the particle in some energy is equal to
one
as always when we carry out our
normalization notice that the
normalization condition
only places a constraint on the
magnitude not the phase
so we're left with a global phase (which
is again ambiguous) out the front
of this wave function but the relative
phase between these two
contributions to the superposition is
important so that would change things
but there's a global complex phase at
the front of this which is arbitrary
because the global phase in quantum
mechanics is unobservable
okay thanks for your time