The Wave Function 04

— 1.5. Momentum and other Dynamical quantities —

Let us suppose that we have a particle that is described by the wave function {\Psi} then the expectation value of its position is (as we saw in The Wave Function 02):

\displaystyle <x>=\int_{-\infty}^{+\infty}x|\Psi(x,t)|^2\, dx

Neophytes interpret the previous equations as if it was saying that the expectation value coincides with the average of various measurements of the position of a particle that is described by {\Psi}. This interpretation is wrong since the first measurement will make the wave function collapse to the value that is actually obtained and if the following measurements of the position are done right away they’ll just be of the same value of the first measurement.

Actually {<x>} is the average of position measurements of particles that are all described by the state {\Psi}. That is to say that we have two ways of actually accomplishing what is implied by the previous interpretation of {<x>}:

  1. We have a single particle. Then after a position measurement is made we have to able to make the particle to return to its {\Psi} state before we make a new measurement.
  2. We have a collection – a statistical ensemble is a more respectable name – of a great number of particles (in order for it to be statistically significant) and we arrange them all to be in state {\Psi}. If we perform the measurement of the position of all this particles, then average of the measurements should be {<x>}.

To put it more succinctly:

The expectation value is the average of repeated measurements on an ensemble of identically prepared systems.

Since {\Psi} is a time dependent mathematical object it is obvious that {<x>} also is a time dependent quantity:

{\begin{aligned} \dfrac{d<x>}{dt}&= \int_{-\infty}^{+\infty}x\dfrac{\partial}{\partial t}|\Psi|^2\, dx \\ &= \dfrac{i\hbar}{2m}\int_{-\infty}^{+\infty}x\dfrac{\partial}{\partial x}\left( \Psi^*\dfrac{\partial \Psi}{\partial x}-\dfrac{\partial \Psi^*}{\partial x}\Psi \right)\, dx \\ &= -\dfrac{i\hbar}{2m}\int_{-\infty}^{+\infty}\left( \Psi^*\dfrac{\partial \Psi}{\partial x}-\dfrac{\partial \Psi^*}{\partial x}\Psi \right)\,dx \\ &= -\dfrac{i\hbar}{m}\int_{-\infty}^{+\infty}\left( \Psi^*\dfrac{\partial \Psi}{\partial x}\right)\,dx \end{aligned}}

where we have used integration by parts and the fact that the wave function has to be square integrable which is to say that the function is vanishingly small as {x} approaches infinity.

(Allow me to go on a tangent here but I just want to say that rigorously speaking the Hilbert space isn’t the best mathematical space to construct the mathematical formalism of quantum mechanics. The problem with the Hilbert space approach to quantum mechanics is two fold:

  1. the functions that are in Hilbert space are necessarily square integrable. The problem is that many times we need to calculate quantities that depend not on a given function but on its derivative (for example), but just because a function is square integrable it doesn’t mean that its derivative also is. Hence we don’t have any mathematical guarantee that most of the integrals that we are computing actually converge.
  2. The second problem is that when we are dealing with continuous spectra (later on we’ll see what this means) the eigenfunctions (we’ll see what this means) are divergent

The proper way of doing quantum mechanics is by using rigged Hilbert spaces. A good first introduction to rigged Hilbert spaces and their use in Quantum Mechanics is given by Rafael de la Madrid in the article The role of the rigged Hilbert space in Quantum Mechanics )

The previous equation doesn’t express the average velocity of a quantum particle. In our construction of quantum mechanic nothing allows us to talk about the velocity of particle. In fact we don’t even know what the meaning of

velocity of a particle

is in quantum mechanics!

Since a particle doesn’t have a definitive position prior to is measurement it also can’t have a well defined velocity. Later on we’ll see how how to construct the probability density for velocity in the state {\Psi}.

For the purposes of the present section we’ll just postulate that the expectation value of the velocity is equal to the time derivative of the expectation value of position.

\displaystyle  <v>=\dfrac{d<x>}{dt} \ \ \ \ \ (24)

As we saw in the lagrangian formalism and in the hamiltonian formalism posts of our blog it is more customary (since it is more powerful) to work with momentum instead of velocity. Since {p=mv} the relevant equation for momentum is;

\displaystyle  <p>=m\dfrac{d<x>}{dt}=-i\hbar\int_{-\infty}^{+\infty}\left( \Psi^*\dfrac{\partial \Psi}{\partial x}\right)\,dx \ \ \ \ \ (25)

Since {x} represents the position operator operator we can say in an analogous way that

\displaystyle \frac{\hbar}{i}\frac{\partial}{\partial x}

represents the momentum operator. A way to see why this definition makes sense is to rewrite the definition of the expectation value of the position

\displaystyle  <x>=\int \Psi^* x \Psi \, dx

and to rewrite equation 25 in a more compelling way

\displaystyle  <p> = \int \Psi^*\left( \frac{\hbar}{i}\frac{\partial}{\partial x} \right) \Psi \, dx

After knowing how to calculate the expectation value of these two dynamical quantities the question now is how can one calculate the expectation value of other dynamical quantities of interest?

The thing is that all dynamical quantities can be expressed as functions of of {x} and {p}. Taking this into account one just has to write the appropriate function of the quantity of interest in terms of {p} and {x} and then calculate the expectation value.

In a more formal (hence more respectable) way the equation for the expectation value of the dynamical quantity {Q=Q(x,p)} is

\displaystyle  <Q(x,p)>=\int\Psi^*Q\left( x,\frac{\hbar}{i}\frac{\partial}{\partial x} \right)\Psi\, dx \ \ \ \ \ (26)

As an example let us look into what would be the relevant expression for the kinetic energy the relevant definition can be found at Newtonian Mechanics 01. Henceforth we’ll use {T} to denote the kinetic energy instead of {K} in order to use the same notation that is used in Introduction to Quantum Mechanics (2nd Edition).

\displaystyle T=\frac{1}{2}mv^2=\frac{p^2}{2m}

Hence the expectation value is

\displaystyle  <T>=-\frac{\hbar ^2}{2m}\int\Psi^*\frac{\partial ^2\Psi}{\partial x^2}\, dx \ \ \ \ \ (27)

Exercise 3 Why can’t you do integration by parts directly on

\displaystyle  \frac{d<x>}{dt}=\int x\frac{\partial}{\partial t}|\Psi|^2 \, dx

pull the time derivative over onto {x}, note that {\partial x/\partial t=0} and conclude that {d<x>/dt=0}?

Because integration by parts can only be used when the differentiation and integration are done with the same variable.

Exercise 4 Calculate

\displaystyle \frac{d<p>}{dt}

First lets us remember the the Schroedinger equation:

\displaystyle  \frac{\partial \Psi}{\partial t}=\frac{i\hbar}{2m}\frac{\partial^2\Psi}{\partial x^2}-\frac{i}{\hbar}V\Psi \ \ \ \ \ (28)

And its complex conjugate

\displaystyle  \frac{\partial \Psi^*}{\partial t}=-\frac{i\hbar}{2m}\frac{\partial^2\Psi^*}{\partial x^2}+\frac{i}{\hbar}V\Psi^* \ \ \ \ \ (29)

for the time evolution of the expectation value of momentum is

{\begin{aligned} \dfrac{d<p>}{dt} &= \dfrac{d}{dt}\int\Psi ^* \dfrac{\hbar}{i}\dfrac{\partial \Psi}{\partial x}\, dx\\ &= \dfrac{\hbar}{i}\int \dfrac{\partial}{\partial t}\left( \Psi ^* \dfrac{\partial \Psi}{\partial x}\right)\, dx\\ &= \dfrac{\hbar}{i}\int\left( \dfrac{\partial \Psi^*}{\partial t}\dfrac{\partial \Psi}{\partial x}+\Psi^* \dfrac{\partial}{\partial x}\dfrac{\partial \Psi}{\partial t} \right) \, dx\\ &= \dfrac{\hbar}{i}\int \left[ \left( -\dfrac{i\hbar}{2m}\dfrac{\partial^2\Psi^*}{\partial x^2}+\dfrac{i}{\hbar}V\Psi^* \right)\dfrac{\partial \Psi}{\partial x} + \Psi^*\dfrac{\partial}{\partial x}\left( \dfrac{i\hbar}{2m}\dfrac{\partial^2\Psi}{\partial x^2}-\dfrac{i}{\hbar}V\Psi \right)\right]\, dx\\\ &= \dfrac{\hbar}{i}\int \left[ -\dfrac{i\hbar}{2m}\left(\dfrac{\partial^2\Psi^*}{\partial x^2}\dfrac{\partial\Psi}{\partial x}-\Psi^*\dfrac{\partial ^3 \Psi}{\partial x^3} \right)+\dfrac{i}{\hbar}\left( V\Psi ^*\dfrac{\partial\Psi}{\partial x}-\Psi ^*\dfrac{\partial (V\Psi)}{\partial x}\right)\right]\, dx \end{aligned}}

First we’ll calculate the first term of the integral (ignoring the constant factors) doing integration by parts (remember that the boundary terms are vanishing) two times

{\begin{aligned} \int \left(\dfrac{\partial^2\Psi^*}{\partial x^2}\dfrac{\partial\Psi}{\partial x}-\Psi^*\dfrac{\partial ^3 \Psi}{\partial x^3}\right)\, dx &= \left[ \dfrac{\partial \Psi^*}{\partial x^2} \dfrac{\partial \Psi}{\partial x}\right]-\int\dfrac{\partial \Psi^*}{\partial x}\dfrac{\partial ^2 \Psi}{\partial x^2}\, dx- \int \Psi^*\dfrac{\partial ^3 \Psi}{\partial x^3}\, dx \\ &=-\left[ \Psi ^*\dfrac{\partial ^2 \Psi}{\partial x^2} \right]+\int \Psi^*\dfrac{\partial ^3 \Psi}{\partial x^3}\, dx - \int \Psi^*\dfrac{\partial ^3 \Psi}{\partial x^3}\, dx \\ &= 0 \end{aligned}}

Then we’ll calculate the second term of the integral

{\begin{aligned} \int \left( V\Psi ^*\dfrac{\partial\Psi}{\partial x}-\Psi ^*\dfrac{\partial (V\Psi)}{\partial x} \right)\, dx &= \int \left( V\Psi ^*\dfrac{\partial\Psi}{\partial x}-\Psi ^* \dfrac{\partial V}{\partial x}\Psi-\Psi ^*V\dfrac{\partial \Psi}{\partial x} \right)\, dx\\ &= -\int\Psi ^* \dfrac{\partial V}{\partial x}\Psi\, dx\\ &=<-\dfrac{\partial V}{\partial x}> \end{aligned}}

In conclusion it is

\displaystyle  \frac{d<p>}{dt}=<-\dfrac{\partial V}{\partial x}> \ \ \ \ \ (30)

Hence the expectation value of the momentum operator obeys Newton’s Second Axiom. The previous result can be generalized and its generalization is known in the Quantum Mechanics literature as Ehrenfest’s theorem


Quantum Mechanics Introduction

— 1. Motivation —

Unlike we did so far in the previous posts first we’ll begin to study Quantum Mechanics by mentioning some of the experiments that motivated the reformulation of Classical Physics into the body of knowledge that is Quantum Mechanics.

Also we’ll spend more time explaining our initial formulations.

— 1.1. New results, new conceptions —

Anyone that has ever realized an experiment before knows that if one wants to know anything about a system that is being studied one needs to interact with it. In a more formal language one should say: “The act of measuring a system introduces a disturbance in the system”.

Thus far we’ve used the concept of mechanical state on our analysis of physical systems. A more critical eye to the concept of reveals the following:

  1. In principle the disturbance can, in some cases, be made as small as one wants. The fact that exists a limit in our measuring apparatus lies in the nature of the apparatus and not in the nature of the theory being used.
  2. There exists some disturbances whose effect can’t be neglected. Nevertheless one can always calculate exactly the value of the effect of the disturbance and compensate it in the value of the quantity that is being measured afterwards.

Thus one can say that our very successful Classical Theory of Physics is casual and deterministic.

Despite its many successes our Classical Theory of Physics had a few dark clouds hovering over it:

The persistence of these experimental results and the failure of accommodating them in the classical world view indicated that a revision of concepts was needed:

  • Corpuscular entities were demonstrating wavelike behavior.
  • Wave entities were demonstrating pointlike behavior.
  • There exists a statistical nature in atomic and subatomic phenomena that seems to be an essential characteristic of Nature.
  • Nature’s atomic nature (no pun intended) implies that the process of measure has to be thought over: some disturbances can’t be made arbitrarily small since there exists a natural limit to how small a disturbance can be made.

— 1.2. Double-slit experiment —

In order to make our previous discussion more concrete we’ll look into a particular experiment that makes it clear how disparate are the two world views that we’ve been discussing.

— 1.2.1. Double-slit and particles —

Suppose an experimental apparatus which consists of a wall with two opening and a second wall that will serve as target. A beam of particles is fired upon the first wall. Some of those particles will hit the wall and of some will pass through the slits and hit the second wall.

Double slit experiment with particles

Double slit experiment with particles

The particles that pass through slit {1} are responsible for the probability curve {P_1} while the

particles that pass through slit {2} are responsible for the probability curve {P_2}. The resulting probability curve, {P_{12}}, results from the algebraic sum of the curves {P_1} and {P_2}.

— 1.2.2. Double-slit and waves —

Now if we direct a beam of waves on the same experimental apparatus the pattern one observes in the second wall is:

Double slit experiment with waves

Double slit experiment with waves

Now what we’ll study is the intensity curve. The intensity curve {I_1} is the result one would observe if only slit {1} was open, while the intensity curve {I_2} is what one would observe if only slit {2} was open.

The resulting intensity is given by the mathematical expression {I_{12}=|h_1+h_2|^2= I_1+2I_1I_2 \cos \theta}. The last term is responsible for the interaction between the waves that come from slit {1} with the waves that come from slit {2} which ultimately is what causes the interference pattern one observes in the second wall.

— 1.2.3. Double slit and electrons —

Now that we familiarized ourselves with the behavior of particles and waves under the double slit we’ll study what one observes when we subject an electron beam through the same apparatus.

Experimental results show conclusively that electrons are particles so we’re expecting to see the same pattern that we saw for macroscopic particles.

Nevertheless this is what Nature has for us!:

Double slit experiment with electrons

Double slit experiment with electrons

In the case of electrons Nature forces to think again in terms of curves related to wave phenomena. But unlike macroscopic wave phenomena this aren’t curves of intensity but they are curves of probability. This introduces an apparent oxymoron since probability curves are a concept that is intrinsic to particles while an interference pattern is intrinsic to waves…

In order to explain what we’re observing one has to assume that each probability curve {P_i} is associated to a probability amplitude {\phi_i}. In order to calculate the probability one has to calculate the modulus squared of the probability amplitude, {P_i=\phi_i^2}. Thus one has to first calculate the sum of probability amplitudes of an electron passing through slit {1} or passing through slit {2} and only then should we calculate the modulus squared of the probability amplitude of an electron passing through slit {1} or slit {2}:

\displaystyle P_{12}=|\phi_1+\phi_2|^2

— 2. Basic concepts and preliminary definitions —

After our initial discussion I suppose that you understand why physicists had the need to introduce a new paradigm that made sense of what’s happening in the atomic and sub- atomic level. Now it’s time to introduce our usual initial definitions.

Definition 1 An operator is a mathematical operations that carries a given function into another function.

As an example of an operator we have {\dfrac{d}{dx}} which a transforms a functions into its {x} derivative. As another example of an operator we have {2\times} that transforms a function into its double.

Definition 2 The linear momentum is represented by the operator

\displaystyle p=\dfrac{\hbar}{i}\dfrac{d}{dx} \ \ \ \ \ (1)

Definition 3 The energy of a particle is represented by the operator:

\displaystyle E=i\hbar\dfrac{d}{dt} \ \ \ \ \ (2)

Definition 4 For a free particle the following mathematical relationships are valid:

{\begin{aligned} \label{eq:relacoesdebroglie} k &= \frac{\hbar}{p}\\ \omega &= \frac{E}{\hbar} \end{aligned}}

— 3. Quantum Mechanics Axioms —

After this batch of initial definitions that allows us to know the entities that will take part in the construction of our quantum vision of the world it is time for us to define the rules that govern them

Axiom 1 The quantum state is defined by the specification of the relevant physical quantities and is represented by a complex function {\Psi(x,t)}
Axiom 2 To every observable in Classical Physics ({A}) there corresponds a linear, Hermitian operator ({\hat{A}}) in quantum mechanics.
Axiom 3 Measurements made to an observable associated with the operator {\hat{A}} done on a quantum system specified by {\Psi(x,t)} will always return {a} where {a} is an eigenvalue of {a}.

\displaystyle \hat{A}\Psi(x,t)=a\Psi(x,t) \ \ \ \ \ (3)

Axiom 4 The probability that a particle is in the volume element {dx} is denoted by{P(x)dx} and is calculated by

\displaystyle P(x)dx=|\Psi(x,t)|^2dx \ \ \ \ \ (4)

Axiom 5 The average value of a physical quantity {A}, represented by {<A>}, is

\displaystyle <A> = \int \Psi^*\hat{A}\Psi \ \ \ \ \ (5)

Where the integral is calculated in the relevant volume region.

Axiom 6 The state of a quantum system evolves according to the Schrodinger Equation.

\displaystyle \hat{H}\Psi= i\hbar\frac{\partial \Psi}{\partial t} \ \ \ \ \ (6)

Where{\hat{H}} is the hamiltonian operator and corresponds to the total energy of a quantum system.