Download Today`s class: Schrödinger`s Cat Paradox

Today’s class:
•  Finish our discussion on
Schrödinger's cat and quantum
computers
•  Schrödinger in 3D
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’. (Cat
emits black-body radiation and interacts with air molecules
around her ! poor kitty gets cold – if you look or not!)
Biggest systems realized in labs to date are typically only a
few, extremely well isolated atoms (not quite a whole cat).
Schrödinger’s Cat Paradox
An atom has a
50/50 probability
of undergoing
alpha decay in
the time t½.
Put a cat in a box
with this radioactive atom, and
a very sensitive
detector. If an α
•  After t½, the atom’s wave function is given by:
particle is
detected, a
hammer breaks a
box of cyanide
and the cat dies. •  So the cat’s wave function is given by:
Does the observer kill the cat??
Looking at cat = measurement.
•  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. In fact, a
measurement is any interaction with the
environment – intentional or not)
•  Cat emits black-body radiation and interacts with
air molecules around her. This are ‘unintentional
measurements’!
•  “Measurement” process does not require a
conscious observer!
“Cat states”
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). But still very useful ! quantum computer
“Decoherence”
•  Decoherence = (typically unintentional) “measurement”
caused by interactions with environment.
•  Cat paradox resolved: wave function of cat is
“measured” by air-molecules & black-body radiation. So
cat is either dead or alive, but 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 (c0Ψ0 + c1Ψ1 ).
•  This means you can manipulate both states at
once (“parallel processing”).
Quantum Computing
•  Classical computer – N bits (2N states)
–  can only do N calc at once - linear processing!
•  Quantum computer – N qbits can be in a
superposition of states
–  can do 2N calculations at once (64 qbits: ~2*1019
operations 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
Quantum Computing
•  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 (Shor’s algorithm)
–  A few other obscure problems
•  If you could factor large prime numbers,
you could decode all encrypted
information in the world in seconds!
(Guess who funds this research…)
Chapter 8. The 3D Schrodinger Equation
That’s all about superposition states.
In 1D:
Next:
Schrödinger in 2D and 3D
! Hydrogen atom
In 2D:
In 3D:
3D example: “Particle in a rigid box”
−
 2 $ ∂2
∂2 ∂2 '
& 2 + 2 + 2 )ψ (x, y,z) + V (x, y,z)ψ (x, y,z) = Eψ (x, y,z)
2m % ∂x ∂y ∂z (
ψ(x,y,z) = X(x)Y(y)Z(z)
What is
∂2
ψ (x, y,z) ?
∂x 2
€
Simplest case: 3D rigid box:
V(x,y,z) = 0 inside; ∞ outside.
€
c
Use mathematics of separation of variables
(does not always work, but it works here):
a
b
Assume we could write the solution as:
ψ(x,y,z) = X(x)Y(y)Z(z)
"separated function"
a) X”Y”Z”
b) X”YZ + XY”Z”
c) X”YZ
d) None of the above
Plug it in the Schrödinger eqn. and see what happens!
ψ(x,y,z) = X(x)Y(y)Z(z) Now, calculate the derivatives for each
coordinate:
(Do the same for y and z parts)
with: ψ(x,y,z) = X(x)Y(y)Z(z)
Now put in 3D Schrödinger and see what happens:
2
− ( X"YZ +XY"Z + XYZ" ) + V (x, y,z)(XYZ) = E(XYZ)
2m
(For simplicity I wrote X instead of X(x) and X" instead of
Divide both sides by XYZ
€
So we re-wrote the Schrödinger equation as:
For the particle in the box we said that V=0 inside and V=∞
outside the box. Therefore, we can write:
)
for the particle inside the rigid box.
If we call this const. '-kx2' we can write:
X"(x) = - kx2 X(x)
The right side is a constant (i.e. it does
not depend on x, y or z):
A) True
Does this look familiar?
B) False
How about this:
ψ"(x) = - k2 ψ(x)
(and similar for Y and Z)
! This is the Schrödinger equation for a particle in a onedimensional rigid box!! We already know the solutions for
this equation:
The right side is independent from x! ! left side must
be independent from x as well!!
!
Repeat for Y and Z:
2D box: Square of the wave function for nx=ny=1
‘Percent’ relative
to maximum
And:
And the total energy is:
or:
with:
, if a=b=c
“cube”
Now, remember:
ψ(x,y,z) = X(x)Y(y)Z(z)
Done!
2D box: Square of the wave function of selected
excited states
100%
0%
Degeneracy
Sometimes, there are several solutions with the exact
same energy. Such solutions are called ‘degenerate’.
E = E0(nx2+ny2+nz2)
ny
nx
Degeneracy of 1 means “non-degenerate”
The ground state energy of the 2D box of size L x L is 2E0, where
E0 = π2ħ2/(2mL2) is the ground state energy of a 1D box of size L.
y
L
E=E0(nx2+ny2)
L
x
What is the energy of the 1st excited state of this 2D box?
a) 3E0
b) 4E0
c) 5E0
d) 8E0
nx=1, ny=2 or nx=2 ny=1