Physics Notes: Principles of Elementary Quantum Mechanics

A formatting note: this page is, in many ways, an experiment. Instead of presenting equations using GIF images, I used some advanced HTML tricks. Consequently, the page may not appear correctly in all browsers. I routinely test pages using Internet Explorer and Mozilla Firefox on Windows, and occasionally with Netscape 7.1 on Windows and Linux. Your mileage with other browsers/platforms may vary. Needless to say... I can't wait for MathML!

State

A state is something that a system is in. The system is where I perform my measurement; the state is the result of that measurement.

When I perform one kind of a measurement, and then I perform another kind of a measurement, the two results are correlated. In particular, the first result may determine the probability that the second measurement will yield a specific value, i.e., that the system will be in a specific state with respect to the second measurement, as opposed to some other state.

The abstract symbol for a state x is | x>. This is just a label; it is not a number.

The symbol for the transition from state x to state y is <y | x>. This, actually, is a number!

Amplitude

Specifically, it is a complex number called the amplitude. The reason why it is a complex number is experimental: we found that if there are two possible ways for a system to reach state y starting from state x, it is not the probabilities, but these complex numbers that will need to be summed:

<y | x> = <y | x>first route + <y | x>second route <assumption>

Probability

The actual probability that the system in state x will also be in state y is computed as the square of the absolute value of the complex number:

P(x → y) = |<y | x>|² = <y | x><y | x>* <assumption>

Base states

It is assumed as an axiom that any state can be expressed as a sum of base states:

| x> = 
 		 ∑
i		  | i><i | x>
 
<assumption>

What we know about the base states is that they are orthogonal:

<i | j> = δij

State vector

The contribution of each base state | i> to | x> is characterized by <i | x>, which is just a complex number. The state | x> can, therefore, be viewed as a vector that is expressed in terms of base vectors | i> in some complex vector space.

The transition amplitude from state x to state y can be expressed through a set of base states as:

<y | x> = 
 		 ∑
i		  |<y | i><i | x>
 
(1)

The probability that a system in state x is in state x is unity:

<x | x> = 
 		 ∑
i		  <x | i><i | x> = 1
 

The probability that a system in state x is found in some base state is also unity:

∑
i		  |<i | x>|² = 
 		 ∑
i		  <i | x><i | x>* = 1
 

From this one can see that

<i | x> = <x | i>*

And since any state y can be a base state in some set of base states, it is true in general that

<y | x> = <x | y>*

Operators

When you do something to a system, you change its state. This is expressed by an operator acting on that state:

| y> = Â | x>

This is defined to mean the following:

 | x> = 
 		 ∑
ij		  | i><i | Â | j><j | x>
 

Which means that  is just a collection of matrix elements Aij, expressed with respect to some set of base states.

Expectation value

A measurement may be expressed in the form of an operator. If this is the case, the average (expectation) value of that measurement can be expressed as:

Aav = <x | Â | x>

which really is just shorthand for

Aav = 
 		 ∑
ij		  <x | i><i | Â | j><j | x>
 

Wave function

What if there is an infinite number of states? For instance, a particle's position l along a line may be expressed in terms of the set of individual positions as base states. But there is an infinite number of such positions possible. Thus, our sum

| l> = 
 		 ∑
x		  | x><x | l>
 

becomes instead the integral

∫ | x><x | l>dx

This equation has little practical meaning since | x> is just an abstract symbol. However, the probability that a system in state l is later found in state k, previously expressed as the sum (1):

<k | l> = 
 		 ∑
x		 |<k | x><x | l>
 

is now the integral

∫ <k | x><x | l>dx

Both <k | x> and <x | l> are just complex numbers; complex-valued functions, in fact, of the continuous variable x:

<k | x> = <x | k>* = φ*(x)
<l | x> = ψ(x)

These functions are called wave functions mainly because they typically appear in the form of periodic complex-valued functions. With their help, the transitional probability can now be expressed as:

∫ φ*(x)ψ(x) dx

and the expectation value of an operator can be written as

∫ φ*(x)Âφ(x) dx (2)

Algebraic operator

In this context, Â no longer works as a matrix operator converting a state vector into another state vector, but as an algebraic operator converting a wave function into another wave function. How the matrix operator, expressed in terms of states and amplitudes, and the algebraic operator, expressed usually as a differential operator, relate to each other is another question!

Position operator

If we know the probability P(x) that a particle will be at position x, we can compute the average position of the particle after many measurements as follows:

∫ xP(x) dx

But this is the same as

∫ xφ*(x)φ(x) dx = ∫ φ*(x)xφ(x) dx

which is formally identical to the expectation value (2) for a measurement that can be expressed in the form of an operator x   ˆ. In other words, x   ˆ can be viewed as the position operator. When the base states are positions, the position operator is just a multiplication of the wave function by x.

Momentum operator

The same computation can be performed for the momentum, using as base states states of definite momentum:

∫ pφ*(p)φ(p) dp = ∫ φ*(p)pφ(p) dp

Question is, can the momentum operator be expressed in terms of base states of position?

The amplitude of a system, which is in state β, to be found in a state of definite momentum p, is just the definite integral

∞
∫
–∞		  <p | x><x | β> dx

The relationship between position and momentum, specifically the amplitude for a particle to be found at position x after it has been measured to have momentum p is assumed to be

<x | p> = eipx/ħ <assumption>

So our integral becomes

<p | β> =  ∞
∫
–∞		 e–ipx/ħ<x | β> dx
(3)

Now let's use, as | β>, the state p   ˆ | k> (it can be any state after all) where <x | k> = φ(x) and the k's are assumed to be states of definite momentum. This way, our earlier expression becomes the expectation value of the momentum:

∞
∫
–∞		  <p | x><x | β> dx =  ∞
∫
–∞		 <p | x><x | p   ˆ | k> dx =  ∞
∫
–∞		  <p | x>p<x | k> dx

Then <x | β> is just p<x | k> = pφ(x), and:

<p | β> =  ∞
∫
–∞		  e–ipx/ħpφ(x) dx

The integral can be computed by observing that de–ipx/ħ/dx = (–i/ħ) pe–ipx/ħ, integrating in parts, and assuming that φ(x) = 0 when x = ±∞:

<p | β> = 
ħ
 ∞
∫
–∞		 e–ipx/ħ 
∂φ
  dx
i ∂x

so, from (3):

<x | β> = 
ħ
  
∂φ
i ∂x

and we now have an expression for the momentum operator p   ˆ:

 p   ˆ = 
ħ
  
∂
i ∂x

Time displacement

How does a system evolve over time? Let's consider the time displacement operator Û(t1, t2):

<χ | Û(t1, t2) | φ>

S-matrix

When t1→ –∞ and t2→ +∞, we call Û(t1, t2) the S-matrix.

Making t1 = t and t1 = t + ∆t, observing that when ∆t = 0, Uij (in some coordinate representation) must be δij, and assuming that for small ∆t, the change in φ will be linear, we get:

Uij = δij – 
i
 Hij(t)∆t
ħ

(the factor –i/ħ is introduced for reasons of convenience.)

In other words, the difference between the wave function of the two states can be expressed as:

φ' – φ = – 
i
 ∆t Ĥφ
ħ

or, dividing by ∆t and recognizing the left-hand side as a time differential:

iħ 
∂φ
  = Ĥφ
∂t

Schrödinger equation

Schrödinger, that kind chap, then just decided to use in place of Ĥ an operator that he concocted up on the basis of the classical expression for energy:

E = 
p²
  + V
2m

His equation:

iħ 
∂φ
  = 
–ħ²
  ∇²φ + Vφ
∂t 2m
<assumption>

describes the wave function of a particle moving in a potential field V.

A crucial thought is that the Schrödinger equation is not as fundamental as you might have been led to believe. Indeed, there's no single Schrödinger equation; the actual equation of a system depends on the characteristics of that system, and is often derived heuristically, through the process of operator substitution.

Operator substitutions

One result is a "rule of thumb": substitution rules that are used to derive quantum operators from the classical quantities of momentum, energy, and position:

 p   ˆ → 
ħ
  
∂
i ∂x
Ĥ → iħ 
∂
∂t

x   ˆ → x

Commutativity

The operators x   ˆ and p   ˆ do not commute:

(x   ˆ o p   ˆ)φ = x 
ħ
  
∂φ
i ∂x
(p   ˆ o x   ˆ)φ = 
ħ
  
∂(xφ)
  = 
ħ
  
∂x
  φ + 
ħ
  x 
∂φ
i ∂x i ∂x i ∂x
(p   ˆ o x   ˆ – x   ˆ o p   ˆ)φ = 
ħ
  φ
i

Probability Current

A simple manipulation of the Schrödinger equation – multiplying on the left by φ*, multiplying the equation's complex conjugate on the left by φ, and subtracting one from the other – can lead to the continuity equation:

 φ* ┌
│
│
└
–ħ²
 ∇²φ + Vφ – iħ 
∂φ
 ┐
│
│
┘		 – φ ┌
│
│
└
–ħ²
 ∇²φ* + Vφ* + iħ 
∂φ*
 ┐
│
│
┘		  =
2m ∂t 2m ∂t
–ħ²
 (φ*∇²φ + ∇φ*∇φ – ∇φ∇φ* – φ∇²φ*) – iħ ┌
│
│
└		 φ* 
∂φ
  + φ 
∂φ*
 ┐
│
│
┘		  =
2m ∂t ∂t
–ħ²
  ∇(φ*∇φ – φ∇φ*) – iħ 
∂φ*φ
2m ∂t

or, substituting

j = 
–iħ²
  (φ*∇φ – φ∇φ*)
2m

ρ = ħφ*φ

we get

–i ┌
│
│
└		 ∇j + 
∂ρ
 ┐
│
│
┘		  = 0
∂t
∇j + 
∂ρ
  = 0
∂t

References

Feynman, Richard P., The Feynman Lectures on Physics III., Addison-Wesley, 1977
Aitchison, I. J. R. & Hey, A. J. G., Gauge Theories in Particle Physics, Institute of Physics Publishing, 1996