Today’s class:
Announcements
• More on superposition states
Online survey due Friday (extra credit)
and Schrödinger's cat
This Friday: last graded HW
• ‘Measurements’
Superposition state
Ψ2(x,t)
|Ψn(x,t)|2 = |ψn(x)|2 : time independent
2
But: Ψ ( x, t ) =
2
1
2
Ψ1 ( x, t ) +
1
2
Ψ2 ( x, t )
2
Example of Superposition state
Ψ1(x,t)
Easiest example I can think of: Double-slit experiment:
2
= ψ1( x) + ψ 2 ( x) +ψ1( x)ψ 2 ( x) cos((E2 − E1 ) ⋅ t / h)
1
2
1
2
ψ1
ψ2
Probability density of detecting it at the screen:
|Ψ|2 = |c1Ψ1 + c2Ψ2 |2
0
2
2
= 12 ψ1 + 12 ψ 2 +ψ1ψ 2 cos((E2 − E1 ) ⋅ t / h)
Note: Here, ψ1 and ψ2 have same energy!
(E2-E1)=0. interference pattern has no time dependence
x
Superposition state:
‘Measurements’ in QM
Mathematically easy!
Ψ( x, t ) = c1Ψ1 ( x, t ) + c2 Ψ2 ( x, t ) + ...
with:
∫
∞
−∞
2
Ψ( x, t ) dx = 1 (i.e. only one particle!)
But what does this mean??!
Need to talk about ‘measurements.’
A measurement will tell us where the
particle is or what energy it has.
A measurement returns specific values for one or more
aspects (like position or energy) of a quantum state.
The issue of measurement lies at the heart of the
problem of the interpretation of quantum mechanics,
for which there is currently no consensus!
Let me show you why<
Why a matter of interpretation?
Before measurement:
Electron is spread out
in space.
After measurement:
Electron is localized!
I think that a particle must have a separate reality
independent of the measurements. That is an electron
has spin, location and so forth even when it is not being
measured. I like to think that the moon is there even if I
am not looking at it.
- A. Einstein
Problem: How could it interfere with itself if it wasn’t spread
out in space before the measurement?
Did the measurement cause the localization or was the
particle localized to begin with and we just didn’t know its
position (since we didn’t measure it yet)?
Delocalized (can go
through 2 slits at once)
Interference pattern
(confirmed by measurement)
Measurement
• Measurement is a discontinuous process, not
described by the Schrodinger equation.
(Schrodinger describes everything before and after,
but not moment of the measurement.)
• If you measure energy of particle, will find it in a
state of definite energy (= energy eigenstate).
Einstein: “Always localized”
No interference!!
• If you measure position of particle, will find it in a
state of definite position (= position eigenstate).
• Unlike classical physics, measurement in QM
doesn’t just find something that was
already there – it CHANGES the system!
How to compute the probability of
measuring a particular state:
Suppose you have a particle with wave function
Ψ(x,t) = c1 Ψ1(x,t) + c2 Ψ2(x,t) + c3 Ψ3(x,t) + …
• Measuring position:
Measuring position
• Example:
double slit experiment:
|Ψ(x,t0)|2
b
2
P(a to b) = ∫ Ψ ( x, t ) dx
|Ψ(x,t)|2
a
• Measuring energy:
P(En) = |cn|2
ab
There’s a much more formal way to describe this (“Von
Neumann Postulate”), but let’s leave it at that for now.
• Probability density at
screen looks like:
• Probability of measuring
particle at particular pixel:
b
2
P = ∫ Ψ ( x, t ) dx
a
x
ab
Note on position and energy
measurements:
• Energy eigenstates tend to be spread out in space.
• Position eigenstates tend to be localized in space.
• This is why you can’t know both at the same time (wave
packets vs. plane waves)
• Measuring position messes up energy eigenstate and
vice versa.
Examples of superposition states
Electron 50:50 probability to be either in
the n=1 or in the n=2 state:
Ψ ( x, t ) =
Ψ1 ( x, t ) +
1
2
Ψ2 ( x, t )
y
Electron after moving through double-slit:
Probability density of detecting it on screen:
1
2
Measure Position
1
2
Ψ1 +
1
2
Ψ2
2
Measure Energy
Extreme case: Cat in superposition state between
“cat alive” and “cat dead.” Whaaaat??
Schrodinger Cat Paradox
• An atom has
50/50 probability of
undergoing
alpha decay
in a time t½.
• Put a cat in a
box with this
radio-active
atom, and
very sensitive
detector. If α • After t½, the atom’s wave function is given by:
particle
detected,
Ψatom ( x, t ) = 12 Ψdecayed( x, t ) + 12 Ψnotdecayed( x, t )
hammer
breaks box of • So the cat’s wave function is given by:
cyanide and
Ψcat ( x, t ) = 12 Ψdead ( x, t ) + 12 Ψalive ( x, t )
cat dies.
Schrodinger Cat Paradox
• Everybody knows you can’t really have a cat
that’s both dead and alive at the same time.
• Schrodinger illustrated a problem with QM:
it predicts that cat will be in a superposition state
UNTIL WE MEASURE IT, but doesn’t define
what it means to make a measurement. (You
can think of a measurement as any interaction
with the environment – intentional or not)
• In practice, this wasn’t really a problem: people
make measurements in the lab and after
measurements, no weird superposition states.
• Until recently, “measurement problem”
considered a question of philosophy.
Schrödinger's cat
Nowadays, the expression “Schrödinger's cat” (or just “cat
states”) is a synonym for equal superpositions of two
maximally different quantum states (like “a live cat” and “a
dead cat”).
These states are very difficult to create as the smallest
interaction with the environment leads to ‘decoherence’.
Biggest systems realized in labs to date are typically only a
few, extremely well isolated atoms (not quite a whole cat).
“Decoherence”
• At least part of this problem was resolved in the 1980’s:
• Decoherence = “measurement” caused by interactions
with environment.
• Cat paradox resolved: wave function of atom is
“measured” by detector in box, so it either decays or
doesn’t decay and cat is either dead or alive, not both!
• This happens with or without observer – don’t have to
wait for conscious observer to look in the box!
• Decoherence explains why we normally only see
quantum phenomena in carefully isolated systems (e.g.
physics lab).
• But< once we understand it, we can control it – produce
quantum behavior in larger and larger systems.
Quantum Computing –
application of superposition states
• Classical computers store code in “bits”: two
state systems – state can be “0” or “1”
• There are lots of two state systems in nature:
electron spin, atoms with only one probable
transitions, electrons bound to two nuclei, etc.
• In quantum mechanics, a two state system can
be in two states (Ψ0 or Ψ1) or a superposition of
both states (Ψ0 + Ψ1 ).
• This means you can manipulate both states
using only a single bit – a “qubit”
Quantum Computing
Quantum Computing
• Classical computer – N bits (2N states)
– can only do N calc at once - linear processing!
• Quantum computer – N qubits can be in a
N
superposition of states (superposed 2N states,~ 2 2
– Can do 2N calculations at once
– massively parallel computation.
• Sounds great – what’s the catch?
– Can only access one state at a time, can’t even control
which state you measure!
– Really hard to build – best quantum computer on earth
only has 13 qubits – can factor 15.
That’s all on superposition states
/and all about the 1D Schrödinger equation.
• Useful for problems where you need to know
relationships between multiple solutions to a problem,
don’t need to know values of solutions:
– Searching large databases
– Factoring large prime numbers
– A few other obscure problems
• If you could factor large prime numbers, you could
decode all encrypted information in the world in seconds!
Up next:
Schrödinger in 2D and 3D