Friday, February 21, 2014

The Essential Quantum Mechanics

Mathematical formulation of QM

We will focus our attention on the wavefunction, the operative part is "function", as this object is mathematical in nature, and not to be taken as a real wave. If you keep that in mind, a lot of confusion about QM will dissipate.

(1) We will use the Dirac notation, that is, the wavefunction is a vector V denoted by a "ket", | V >.

(2) A ket can be multiplied by any scalar α, β... denoted by α| V >, β| V >...

Note we can also write α| V > as | αV >, that is, a new vector | V' >.

(3)There is a dual vector, the "bra", denoted by < V |.

(4) Note that < V' | = < αV | = < V | α*, where α* is the complex conjugate of α.

(5) Vectors can also be represented by matrices. In the case of the ket vector, | V >, it will be a column matrix, while the bra < V | will be a row matrix, with its elements as the complex conjugate of every element of the vector | V >. See diagram below for a vector in 4-D.

(6) Any vector | V > in an n-dimensional space can be written as a linear combination of n linearly independent vectors | 1 >, | 2 >...| n >.

(7) A set of n linearly independent vectors in an n-dimensional space is called a basis, and so we can write,

| V > = Σ vi | i >,

where i = 1... n, vi are the components of the vector, and the | i >'s are the basis vectors.

(8) We can construct an inner product between two vectors. The analog of the dot product between two vectors V and W, VW = VW cosθ, is < V | W >. In diagram 1, we have,

< V | V > = v1v1* + v2v2* + v3v3* + v4v4* = |V|2

(9) Note that

< V | W > = < W | V >* = Σ Σ vi* wj < i | j >.

(10) If the basis vectors have unit length, then they form an orthonormal base. From (7) above, we can find the jth component of the vector | V > as such,

| V > = Σ vi | i >,

Multiply both sides by < j |

< j | V > = Σ vi< j | i >
= Σ viδij - using (6)
= vj
An important result is that,

vi = < i | V >, and (7) can now be written as,

| V > = Σ | i >< i | V > .

(11) An operator Ω is an instruction to transform a vector | V > into another vector| V' >. This can be written as,

Ω| V > = | ΩV > = | V' >.

Note that Ω acts on the right.

(12) Operators are said to be linear if they obey the following rules: for any α and β,

(12a) Ωα| V > = αΩ| V >

(12b) Ω{α| V > + β| V' >}= αΩ| V > + βΩ| V' >

The action of two successive operators Ω and Λ in general will not be commutative. We designate the commutator as such:

(13) ΩΛ - ΛΩ ≡ [Ω,Λ]

(14) Recall that | V > = Σ | i >< i | V >, we can define the projector operator as,

Pi = | i >< i |

The importance of this operator is that,

I = Σ Pi, where I is the identity operator.

(15) In regard to the dual vector, the bra < V |, we define the operator acting on it as,

< V' | = < ΩV | = < V |Ω, where Ω is called the adjoint operator.

Note that Ω acts on the left.

Also, < V' | = < αV | = < V |α*

Therefore, < V |Ω = < V |α*

In a given basis, the adjoint operation is the same as taking the transpose conjugate.

(16) An important theorem is,

(ΩΛ) = ΛΩ

(17) An operator is said to be Hermitian if,

H = H

(18) An operator is said to be unitary if,

UU = UU = I, where I is the identity operator.

Note that U = U-1, where U-1 is the inverse of U.

An important theorem is that a unitary operator preserves the inner product:

Proof: consider | V' > = U| V > and | W' > = U| W >

Then < W' | V' > = < W |UU | V > = < W | V >

(19) An important problem involves the situation when,

Ω| V > = ω| V >, where ω is a number, real or complex.

We say that the operator Ω rescales the vector | V > by a factor ω. That equation is called an eigenvalue equation: | V > is an eigenket of Ω with eigenvalue ω.

Consider < V |Ω| V > = ω < V | V >, and < V |Ω| V > = ω* < V | V >. If Ω is hermitian, then Ω = Ω. Therefore ω = ω*, and ω is real. Since what we measure are observables, and their measurements must be real numbers, Hermitian operators are perfect candidate for observables in QM.

(20) So far we have dealt with vectors. Now we want to bring in continous functions into our formalism as these will play an important role. We take a function f(x) along a certain interval between 0 and L. Divide this into equal parts, say n=20 parts. Let x=L/20, 2L/20...19L/20. See fig. 2 below.

We denote the ket | fn(x) > as the discrete approximation of f(x).

The basis vector in this space are:

We have,

< xi | xj > = δij

Σ | xi >< xi | = I

| fn(x) > = Σfn(xi) | xi > , where i = 1....n

(21) All we have to do to go from the discrete to the continuous spectrum is,

n → ∞, and Σ → ∫. For instance,

Σ | xi >< xi | = I → ∫ | x >< x | dx = I

(22) Note that < x | f > is just the projection of | f > along the basis | x >, which is just f(x).

< x | f > = f(x)

Likewise, < g | x > = g*(x).

(23) The inner product - see equation (9) - becomes,

< f | g > = ∫ < f | x >< x | g > dx - using (21)

= ∫ f*(x)g(x)dx - using (22)
Particle on a line

Consider a particle moving along a line in the x-direction. Every point on the line can be represented by a function of x, denoted by |Ψ(x)>. The observable would be the operator that locates the particle on the x-axis, say X. In this case we can write,

(24) X|Ψ(x)> = xΨ(x)

The operator X simply multiplies the function Ψ(x) by x. The next step is to find if there are eigenvalues, and what are they. To do that we would need to find the eigenvectors such that,

(25) X | λ > = λ | λ >, where | λ > is the eigenket, and λ is the eigenvalue.

Substituting and transposing,

(26)(x – λ) | λ > = 0

This tells us that whenever x ≠ λ, then the function | λ > is zero. The only place where | λ > is not zero is when x = λ. Let’s plot what | λ > looks like.

As the interval ε decreases to zero at x = λ, the function goes to infinity. This function is known as the Dirac delta function.

(27)| λ > = δ (x – λ)

Note: the area under the delta function is 1 (ε x 1/ε ).

(28) The probability of detecting the particle at position x is,

P(x) = |< x | Ψ >|2 = < x | Ψ >< Ψ | x > = Ψ(x)*Ψ(x)

The implication is that if Ψ(x) is any function of x, we can calculate the probability of finding a particle at x with the above mathematical structure we have constructed.

Momentum

We are going to consider another operator, the differential ∂x, which would give another function of x, ∂xΨ(x).

(29) To be an observable, this operator needs to be Hermitian, which it isn’t in that form. However, K = – i∂x is Hermitian.

Proof:

We need to show that < Ψ | K | Ψ > is real , that is,

< Ψ | K | Ψ > = < Ψ | K |Ψ >*

(i) By definition,

< Ψ | K | Ψ > = ∫ Ψ*(x)(– i∂x )Ψ(x) dx
= – i ∫ Ψ*(x) (∂Ψ(x)/∂x) dx
(ii) Integrating by parts,

< Ψ | K | Ψ > = i ∫ Ψ(x)(∂Ψ*(x)/∂x) dx

(iii) Complex conjugate the above,

< Ψ | K | Ψ >* = – i ∫ Ψ*(x)(∂Ψ(x)/∂x) dx

Which the same as (i), QED.

Our next step is to find the eigenvectors of this operator K, that is,

(30) – i∂x Ψ(x) = k Ψ(x), where k is a real number

Aside from an arbitrary constant, which we can neglect for our purposes, a solution to that equation is,

(31) Ψ(x) = e ikx

These are the eigenvectors of the operator K , which are exponential functions. Note how these functions behave differently from the eigenfunctions of the position, which were the Dirac delta functions. While the position eigenfunctions are peaks, the eigenfunctions of K extend over all spaces, oscillating everywhere with equal probability.

So what is this operator? We can see from fig 6 that after the wavelength λ, the wavefunction will repeat itself. For each cycle,

(32) kλ = 2π, or λ = 2π/k

(33) Here we appeal to history. De Broglie had hypothesized in the wave/particle duality that every wavelength was inversely proportional to the momentum, p = h/λ, where h is the Planck constant.

Substituting for λ,

(34) p = (h/2π) k = ℏ k

So we can see that our operator K differs from the momentum operator by a factor ℏ. One of the most remarkable result of quantum mechanics is that momentum along the x-axis can be represented by the operator,

(35) px = – iℏ∂x

Finally, let us calculate the commutator [x,p] ( see 13 above). Recall that we are dealing with operators, and by definition, these operate on vectors/functions.

(36) [x,p] → [x,p]f = (xp - px)f

= {x(– iℏ∂x) - (– iℏ∂x)x}f
= x(–iℏ∂xf) + x(iℏ∂xf) +iℏf = iℏf
Therefore, [x,p] = iℏ

These two operators don't commute. We see how the Heisenberg Uncertainty Principle is revealed. If we know the position of the particle with definite precision (the Dirac delta functions), then the eigenvectors of the momentum (the exponential functions) tells us we can’t define its momentum as these wavefunctions are spread out equally all over the space (the x-axis). In this case, when the two operators don't commute, [x,p]= iℏ ≠ 0, we say that the position and momentum of a particle are incompatible observables.

On another note, the mathematical formalism of QM is quite different than classical physics. We have Hermitian operators, representing observables, operating on the wavefunction, which contains the information of the system, such as position, momentum, etc. and with this, we can calculate probabilities. A long historical debate ensued as to whether the wavefunction contains all of the information, or is QM incomplete. So far, after nearly 90 years, no one has come up with a better theory.