#dirac notation

[johnlo101]

[peeterjoot]

Operator matrix element

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Weird dreams

I woke up today having a dream still in my head from the night, but it was a strange one. I was expanding out the Dirac notation representation of an operator in matrix form, but the symbols in the kets were elaborate pictures of Disney princesses that I was drawing with forestry scenery in the background, including little…

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[peeterjoot]

bra-ket manipulation problems

[mathjax]

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Some bra-ket manipulation problems.([1] pr. 1.4)

Using braket logic expand

(a)

\begin{equation}\label{eqn:braketManip:20} \textrm{tr}{X Y} \end{equation}

(b)

\begin{equation}\label{eqn:braketManip:40} (X Y)^\dagger \end{equation}

(c)

\begin{equation}\label{eqn:braketManip:60} e^{i f(A)}, \end{equation}

where \( A \) is Hermitian…

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[physicsandchemistryrevision]

Quantum Physics

Dirac notation: Analogy with Cartesian vectors

Cont'd from "Dirac notation: Introduction"

Analogies can be made between the Hilbert space we use in quantum mechanics and the standard 3-dimensionally real Euclidean (or Cartesian) space that we use in classical physics – such as in kinematics wherein we have vectors with components in the x-, y- and z- directions. By defining in which directions the corresponding unit vectors ex, ey and ez act, we can know an awful lot about the state of the vector in that space by projecting it onto those unit vectors. 

Consider a general vector V in ℝ3 (i.e. 3-dimensional, real) Euclidean space using Cartesian co-ordinates.

The projection of our vector onto each axis is given by the 3 Cartesian components of the vector; Vx, Vy and Vz. We’ll discuss later how these axes are defined and, in turn, how a vector can be projected onto it.

Now, if we define our Cartesian unit vectors by the basis state

where

(note that each unit vector is orthogonal to one another since their inner product is zero) and define our vector by its three components

we can find each of our vector’s components by projecting its state onto the x, y, z basis (i.e. the basis vector given by ⟨ e |). So,

This gives us the probability amplitude of finding the vector V in the Cartesian plane.

Thus, by multiplying these matrices together we find our vector in the representation given by e 

(we'll call this equation 5), as was expected!

This is the vector explicitly in Cartesian space, formed by projecting our vector onto another set of vectors. This projection was achieved using the inner product of V with e. As such, we can infer that any vectorial resolving we’ve met before is a simplified inner product. Let’s quickly examine this.

Let’s set up the inner product of V with one of the unitary axial components of e, say ex for instance. In regular vector notation, i.e. to better examine the scalar product, we have

Now, we know that the unitary vector ex has a length of one by definition. Therefore,      

which is often the equation we use to find a certain vectorial component when resolving vectors, depending between which axis we have defined θ. Therefore, we can infer

We can examine the projection of this vector onto every component of ⟨ e | individually to understand how the vector acts in that particular direction. We can either do this by re-defining ⟨ e | V ⟩ = V(e) and projecting this new function onto the individual components of e or by doing this implicitly within Dirac notation. Since we know that we’re dealing with discrete quantities here, let’s use the latter for now.

Let’s first start by continuing the analogy with Dirac notation and represent the state |e⟩ as a superposition of all the possible states within it:

which makes its basis vector

since, in general, | ψ ⟩* = ⟨ ψ | and where by the definition of the e-components,

(recall the conjugate transpose discussed above).

Take equation 5 and multiply through by the particular axis (or representation) we’re interested in. Take the x-component first, ⟨ ex | and project ⟨ e | V ⟩ onto it:

Using the above definitions of components of e,

and since ⟨ ex | = (1  0  0),

and so, by multiplying the matrices we find that 

which is exactly the same result we found above while performing the scalar product between V and ex. Thus,

as a consequence.

The same can be done for all components of the bases represented by ⟨ e |.

We could also ‘cut out the middle-man’ and instead find the direct projection of the vector V onto one of the axes.

and so on, to find

Thus, the above diagram in Dirac’s notation becomes:

The two are completely analagous but come from a different approach. Notice how the vector “state” in Dirac notation (shown in the pink square) has no initial representation until projected onto the representation given by e.

Using this analogy of Cartesian vectors in Dirac notation can help us to understand some of the similarities and differences between Euclidean space and Hilbert space, which we will look at in a future post.

[me-ire-y-no-volvere]

(∂ + m) ψ = 0

[clevermonkey93]

The day will come when I will understand quantum mechanics 3 -- But it is not this day.

me. right now. crying.