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The Complex Vector Space

The Complex Vector Space


In the postulates of quantum mechanics post, we established that the states of a system are described by vectors in the complex vector space.

Quantum mechanics uses the complex vector space as it allows us to do operations that would not have been possible using set theory (classical mechanics) or even using normal vectors with no complex parts.

The vectors we use can be ket vectors - |A>
or bra vectors <A|.

If we were working in a 1D complex vector space (the complex numbers) we could have any complex number z and write it's complex conjugate as z*.
With the bra ket notation we can write a vector as |Z> and then its complex conjugate will be <Z|.

The convention is that ket vectors are written as column vectors and bra vectors are written as row vectors.

Inner Product


One operation you can perform in a complex vector is the inner product and this the equivalent of the dot product used with regular vectors.
The operation Bra vector A multiplied by  Ket vector B can be written as
<A|B>
This is called taking the inner product of B with A.

If K is a constant (may be a complex number as we are in the complex vector space)
<A| K|B> = K<A|B>

The inner product of the sum of two vectors works just like regular multiplication so is quite intuitive.
<A| (|B> + |C>) = <A|B> + <A|C>

The following result tells us that if you swap the way you preform the inner product, you get the complex conjugate of performing the inner product the original way.
AB = (A*B*)*
<A|B> = <B|A> *

Inner product of a vector with itself is its own complex conjugate. The only numbers that are their own complex conjugates are real numbers and so <A|A> is real.
<A|A> = <A|A>* 

For functions we can also define the inner product because functions can also be thought of as vectors. If we have two complex functions
ϕ(x) and Ψ(x)
<ϕ(x)|Ψ(x)> = ∫ dx ϕ(x)*Ψ(x)

Basis Vectors


Basis vectors are perpendicular vectors of unit length.
This means that that the inner product of any 2 basic vectors will always be 0 because they are always perpendicular. This is the same as if you were to dot product the unit i vector with the unit j vector you will get 0.
The inner product can be thought of as a measure of a vectors component in a certain direction. If the vectors are perpendicular, the inner product is 0 because one vector has no component in the other's direction. If the vectors (unit vectors) are parallel, the inner product is 1 because the vectors are in the same direction. Anything between suggests that they are somewhere between parallel and perpendicular.

If we have a 3D vector space, then we need 3 perpendicular/orthogonal vectors to be able to describe every vector in the space. We can then normalize them so that they are they are basis vectors.
This means it should be possible to write any 3D vector as the sum of its basis vectors. This is like being able to describe any point in a 3D system if you have 3 perpendicular axis. The point can be described by taking one value from each perpendicular axis

If we have a general vector |v> we can write it as the sum of basis vectors.


This result will be useful when we start looking at systems in quantum mechanics.



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

 The Postulates Of Quantum Mechanics

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


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