There are many parallels between the Yahtzee thought experiment and quantum mechanics. When
the five dice are thrown in the first round the situation is described by a
probability distribution over five possible states:
The states Ψi form a base in a 5-dimensional real linear space and Ψ(1) is a
vector in this space. Ψ is transformed after each round to a new probability distribution
by the transition matrix P.
Ψ(i+1) = P∙Ψ(i)
The probability of
being in a state with five equal dice after n rounds equals the Ψ5-component of
A physical state in quantum mechanics |Ψ> is also described by a vector.
This vector belongs to a complex linear space, a Hilbert space.
A Hilbert space is a complete inner-product linear space. The inner product or scalar product
of a vector space is a function from pairs of vectors to scalars (real or complex), that satisfies:
With an inner product in place the length or norm of a vector is defined as:
The scalar product in 3 is defined as:
<u,v> = |u|∙|v|∙cosθ where
|u|= and θ is the angle between
u and v.
u and v are called orthogonal if <u,v>=0, a generalization of
perpendicular. A set of vectors are orthonormal if they are
mutually orthogonal and of norm one. A space is complete if every
sequence of vanishing separation has a limit point that belongs to
The die-state is a superposition of five basis states Ψi
just as a quantum state is a linear combination of states.
The quantum state space can be of finite or infinite dimension.
The electron’s spin state can be described by a vector |Ψ> =
a|Ψ↑> + b|Ψ↓>
in a two-dimensional space of spin up and down. A particle confined to
movement along a line is described by a complex-valued
wave-function Ψ(x) x∈.
Ψ(x) belongs to an infinite-dimensional function space spanned by wave
functions Ψn(x)=einx. A particle in three dimensions with a
state that changes in time has a complex valued wave function defined on space-time .
The scalar product is given by :
|Ψ(x)|2 is the probability distribution of the particle.
Since the particle must be somewhere, integrating |Ψ(x)|2 from
−∞ to +∞ gives:
This is similar to the die that has to be in one of five states Ψ1
to Ψ5. The coefficients αi are probabilites that add up to one.
The quantum coefficients are complex-valued amplitudes that must be squared
|αi|2 before they can add up to one.
The probability distribution of the die evolves in discrete steps through the
transition matrix, Ψ(i+1) = PΨ(i). The matrix has
columns that add up to one which will keep the total probability equal to one (check this!).
The quantum mechanical evolution of a state from |Ψ(t0)> to
|Ψ(t)> is also linear, given by a unitary operator
|Ψ(t)> = U(t,t0) |Ψ(t0)>.
Unitary matrices satisfy U ∙U=E. The adjoint operator U
is given by interchanging rows and columns and conjugating the elements.
Unitary operators conserve the norm <Ψ(t),Ψ(t)> =
<Ψ(t)|U ∙U|Ψ(t)> =
<Ψ(t0),Ψ(t0)> = 1. The
traditional Schrödinger equation is another way to describe
is the Hamiltonian operator that acts on elements in the Hilbert space, it
is related to the unitary operator. An incremental evolution U(t+dt,t) is given by:
E is the identity operator. The Hamiltonian for a system is constructed by taking the classical energy function of
the system and replacing momentum variables with linear operators
A particle in potential well V(x) with classical energy H=p2/2m+V(x) would be
described by a wave-function satisfying the Schrödinger
A classical particle is confined to the area with positive kinetic energy. The total energy
Etot exceeds the potential energy V(x). Quantum phenomena can be very strange,
the wave function Ψ(x) of a quantum particle has a probability distribution
|Ψ(x)|2 that extends into the forbidden zone. This makes it possible to tunnel
through potential barriers that would stop a classical particle. Quantum tunneling happens every time a radioactive nucleus emits an
All measurable physical quantities are described by self-adjoint operators O = O.
Self-adjoint operators in a Hilbert space have eigenfunctions
O|φn> = λn|φn> that span the space.
|φn> represents a state that always result in the value λn
when O is measured. Self-adjointness ensures that λn is a non-complex number.
An orthonormal set of basis states can be chosen from these eigenfunctions. Every physical state can be
The possible outcomes of measuring O are restricted to the values λn and
the probability of getting λn in a state with a serial expansion
|Ψ> = Σnαn|φn> is
|αn|2. To get the likelihood of λn, project the
state |Ψ> onto the basis vector |φn> and square the amplitude.
Back to Yahtzee! We can introduce an inner product in the space by defining
<Ψi|Ψj> = δij
This makes (Ψ1 , Ψ2 , Ψ3 ,
Ψ4 , Ψ5 ) an ON-base. Every state can be written as
Ψ = ΣαiΨi with Σα=1.
Each basis state represents a certain outcome of the observation
“maximal number of identical dice”. The difference is
that our amplitudes are real numbers that represents probabilities
and the quantum amplitudes are complex and must be squared to get
A quantum mechanical speciality is the measurement process which represents a reduction
of the wave function in a rather mysterious way into one of the
allowed eigenstates of the observable being measured. This
corresponds in the Yahtzee experiment to the fact that we always
observe a definite outcome never a strange probabilistic
The thought experiment with a large number of parallel dice-throwers represents an all
encompassing situation where all outcomes are realized in
proportion to their likelihood. This is a parallel to the many
worlds interpretation of Q.M. where the wave function never
collapses in a measurement process. The world just divides itself
into parallel universes where all different outcomes are
There is another parallel with Quantum Mechanics in the
Yahtzee probability calculations. The calculations can be described as
follows in traditional Q.M notation.
Initial state |Ψ(1)>
Probability of being in state Ψi is given by the projection:
Second state |Ψ(2)> = P|Ψ(1)>
Probability of being in state Ψi is given by :
Third state |Ψ(2)> = P2|Ψ(1)>
Probability of state Ψi is :
And so on.
To calculate the probability of a certain outcome after n throws you sum all
possible histories that lead to this outcome. To calculate the
probability of Yahtzee in three throws there will be 25 terms. The
j,k term in the summation is the probability of the
following road to Yahtzee in three steps:
This resembles the Feynman path integral formulation of Quantum Field Theory. Q.F.T
replaces the classical notion of a single unique history of a
system with a sum, technically a functional integral over an
infinity of possible histories. The particle paths take place in
space-time. Particles travel in both directions in time but
particles going backwards in time can be reinterpreted as
antiparticles going forward in time. Going from A to B via
different particle interaction is represented by different Feynman
diagrams that represent quantum amplitudes. They will be added
before the absolute is taken and squared. Paths representing
different histories seem to coexist and interfere with each other
to create physically observable effects on the outcome of a