The Postulates Of Quantum Mechanics

The Postulates Of Quantum Mechanics

Just like special relativity, quantum mechanics has some basic postulates on which the theory is based off. Special relativity is quite simple as it takes two postulates and can describe a wide range of effects using intuition and basic algebra. Quantum mechanics is much more complicated and abstract in terms of postulates and the mathematics involved. I will begin a series on quantum mechanics similar to the one on relativity where I discuss quantum mechanics with the mathematics added in. Since the mathematics is much harder than relativity some parts may skip proving certain things simply because I don't know how to. This series may be difficult to follow if you are not familiar with complex numbers and vectors. Also it will likely be useful if you know about matrices, linear operators and trigonometry.

After laying the mathematical foundations of the theory, I will discuss ideas position, momentum, polarization, slit interference and the Schrodinger equation.

Postulate 1

The states of a system are described by vectors in the complex vector space.

A coin has two states, heads and tails. In classical mechanics it is logical to use sets to describe the states of the coin and use operations such as union and intersection. Quantum mechanics uses vectors instead to describe states. Why? This is because in quantum mechanics it makes sense to add different states together or multiply states by complex numbers. In classical mechanics you would never think of doing heads + tails or doing heads * 5. Using vectors in the complex vector space simply means we can do funky things to the states of a system which will involve complex numbers.
Quantum mechanics uses bra ket vectors, In the case of a coin the vectors used to describe is possible states would be |H> and |T>. For a die, you may use |1>, |2>, |3>, |4>, |5> and |6>.

The collection of vectors to describe the collection of states of a system will have a certain number of dimensions. The number of dimensions depends on each system,e.g, a coin uses 2D complex vector space and a dice uses a 6D complex vector space.
A particle in space can be at any point in space (assuming space can be divided infinitely) and so the collection of vectors used to describe the possible positions of a particle has infinite dimensions.

The vectors used to describe different states of a system will be perpendicular/orthogonal. For a coin this means its vector space will consist of 2 perpendicular vectors for heads and tails. For a dice, there will be a 6D vector space with 6 perpendicular basis vectors.
In a later post we will go into more detail about the complex vector space but for now take away that we use complex vectors to describe states.

Postulate 2

Things that you observe/measure in a system correspond to an Hermitian operator
Using vectors in quantum mechanics also means that you can use operators on the vectors (these can be thought of as matrices). Operators which are Hermitian are the mathematical representation of a certain observable property of a system.
This means the operator for position is Hermitian and corresponds to measuring the position of a particle. This is also the case for momentum. In a separate post we will go into what operators are and what it means for them to be Hermitian but for now,
Hermitian operator -> something you can measure.

Postulate 3

The possible values of an observable are given by the eigenvalues of the operator.
When an operator acts on a vector, you get a new vector * a number. This means when an Hermitian operator (H) acts on a system. You get a new vector * a value. The possible values correspond to an observable.
(If A is a general state vector representing a system in a state)
H |A> = λ|C> (where λ is an eigenvalue of the operator)

If the Hermitan operator is X (the operator corresponding to position) the eigenvalues you get are the possible positions of the particle.
It is important that the eigenvalues are real because we measure real quantities. We are yet to perform an experiment and record an imaginary or complex number as the result.

Postulate 4

States for which the observable have a definite value are the eigenvectors of H.
The value for this eigenvector is the corresponding eigenvalues.

In the case of a quantum mechanical coin (we will use Q as the Hermitian operator corresponding to seeing heads or tails on the coin), the vectors corresponding to definitely heads |H> and definitely tails |T> are the eigenvectors of the hermitian operator Q. If the coin is definitely heads then it may be represented by +|H>. The means the eigenvalue is +1 and this is the value we measure on the coin. In the case of a coin we might use +1 for heads and -1 for tails.  This value is arbitrary and we could have picked any number we wanted. Here I picked +1 and -1 because if you were to do an experiment involving the flipping of a coin you might record heads by writing +1 and tails by writing -1.
For things like momentum and position however, the eigenvalues should correspond to an observable momentum or location.

Postulate 5

If |n> is an eigenvector
P(A) = <A|n><n|A>

The probability of observing a certain state of a system (we will call this state A) is given by the component of A along the eigenvector n * its complex conjugate.
Note that this is always real as probabilities can't be complex as far as we know.
The component of A along an "eigen axis"  is <n|A> (gives complex number)
P(A) = <n|A> <n|λ|>
= |<n|A>|² (Also the equivalent of squaring the modulus of the component of A along the eigen vector).

In quantum mechanics probabilities are fundamental. If you repeat an experiment again and again you will yield different results each time. Analyzing the results will show you that each result has a probability of occurring. So what does this formula mean?
Remember eigenvectors are states in which the observer able has a definite value.
When you measure a result, the system has a definite state and so the result must correspond to an eigenvector. The probability of seeing this eigenvector is given by the component of A along the eigenvector (A is a vector corresponding to the system being in an arbitrary state). This is similar to looking at the x component and y component of velocities.


In the next few posts I will continue to refer back to these postulates and hopefully you will start to get a better idea of what they actually mean. After we will be able to apply them to situations such as for the polarization of photons and for slit interference.


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Check Out My Other Posts On Quantum Mechanics (link to all posts)

General Relativity vs Quantum Mechanics

The Heisenberg Uncertainty Principle

Spooky Action At A Distance - Why The Universe May Not Be Real

The Birth Of Quantum Mechanics - The Ultraviolet Catastrophe

Schrödinger’s Kittens - The Boundary Between Quantum And Classical Mechanics

Did you see my previous post? Click the link below to check it out

String Theory