Today’s class: •  Finish our discussion on

Schrödinger's cat and quantum computers

•  Schrödinger in 3D

Schrödinger’s Cat Paradox

•  After t½, the atom’s wave function is given by:

•  So the cat’s wave function is given by:

An atom has a 50/50 probability of undergoing alpha decay in the time t½.

Put a cat in a box with this radio-active atom, and a very sensitive detector. If an α particle is detected, a hammer breaks a box of cyanide and the cat dies.

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).

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!

“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.

“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

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…)

Schrödinger in 2D and 3D ! Hydrogen atom

That’s all about superposition states.

Next:

Chapter 8. The 3D Schrodinger Equation

In 1D:

In 3D:

In 2D:

€

−2

2m∂2

∂x 2+∂2

∂y 2+∂2

∂z2$

% &

'

( ) ψ(x,y,z) +V (x,y,z)ψ(x,y,z) = Eψ(x,y,z)

Simplest case: 3D rigid box: V(x,y,z) = 0 inside; ∞ outside.

Use mathematics of separation of variables (does not always work, but it works here):

Assume we could write the solution as: ψ(x,y,z) = X(x)Y(y)Z(z)

Plug it in the Schrödinger eqn. and see what happens! "separated function"

3D example: “Particle in a rigid box”

a b

c

What is

ψ(x,y,z) = X(x)Y(y)Z(z)

€

∂2

∂x 2ψ(x,y,z)

a) X”Y”Z” b) X”YZ + XY”Z” c) X”YZ d) None of the above

?

ψ(x,y,z) = X(x)Y(y)Z(z) Now, calculate the derivatives for each coordinate:

€

−2

2mX"YZ +XY"Z + XYZ" ( ) +V (x,y,z)(XYZ) = E(XYZ)

Divide both sides by XYZ

(Do the same for y and z parts)

(For simplicity I wrote X instead of X(x) and X" instead of )

Now put in 3D Schrödinger and see what happens:

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.

with: ψ(x,y,z) = X(x)Y(y)Z(z)

The right side is a constant (i.e. it does not depend on x, y or z): A) True B) False

(and similar for Y and Z) The right side is independent from x! ! left side must be independent from x as well!!

!

If we call this const. '-kx2' we can write:

X"(x) = - kx2 X(x)

Does this look familiar?

ψ"(x) = - k2 ψ(x) How about this:

! This is the Schrödinger equation for a particle in a one-dimensional rigid box!! We already know the solutions for this equation:

And:

Repeat for Y and Z:

And the total energy is:

Now, remember: ψ(x,y,z) = X(x)Y(y)Z(z)

Done!

or:

with: , if a=b=c “cube”

2D box: Square of the wave function for nx=ny=1

‘Percent’ relative to maximum

2D box: Square of the wave function of selected excited states

100% 0%

nx ny

Degeneracy Sometimes, there are several solutions with the exact same energy. Such solutions are called ‘degenerate’.

E = E0(nx2+ny

2+nz2)

Degeneracy of 1 means “non-degenerate”

a) 3E0 b) 4E0 c) 5E0 d) 8E0

What is the energy of the 1st excited state of this 2D box?

y

L

L

x

E=E0(nx2+ny

2)

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.

nx=1, ny=2 or nx=2 ny=1