Introduction to Quantum Mechanics and Bell’s Inequality

(Spend 30 minutes of your life and explore the abstract quantum science)

Jiajun Wang

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Introduction

Quantum mechanics is a study of some physical phenomena, which usually happens at a

very small scale (sub-atomic scale), that cannot be explained by Classical mechanics.

Some discoveries, as you may have learned in school, had been made at the beginning

of 20th century which led to the establishment of quantum mechanics. Einstein’s

photoelectric effect, which shows that energy of light wave is packed, show that light is

composed of “light quanta” (later called “photons”). Young’s Double Slit Experiment

shows the wave property of light, as they and superposition when they’re shined upon a

double slit. The wave-particle duality is one of the myths in science.

Bell’s inequality, first published in 1964, points out that two principles that we admit in

real life should not coexist (either one of both should be false). These two principles are

i) reality, meaning that microscopic objects have real properties determining the

outcomes of quantum mechanical measurements and ii) locality, meaning that reality in

one location is not influenced by measurements performed simultaneously at a distant

location. This text will introduce you to the science of quantum mechanics and guide

you to understand Bell’s inequality.

Pre-requite knowledge

In this context, the reader is expected to have a background in linear algebra. A course

like Linear Algebra 1 (favourably Linear Algebra 2) is sufficient to understand this text.

Also, a certain background about classical mechanics and possibilities is required in this

text.

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The Stern-Gerlach Experiment

The Stern-Gerlach experiment greatly illustrates some phenomena that can’t be

explained by classical mechanics.

A Stern-Gerlach apparatus is an apparatus that has an inhomogeneous magnetic field

inside (with the gradient ) in one direction (ex: In the z-direction as shown in the

figure). Particles can enter the apparatus and their magnetic moment (if there is any,

notated as ) will interact with the magnetic field of the apparatus, which results into a

force on the particle that pull the particles away from their original path. We could use

this apparatus to measure the magnetic moment of particles in any one direction, or

select particles with certain magnetic moment in any one direction for future use.

According to classical mechanics, the Force in the z-direction can be calculated from the

following equation:

, where is the force in the z-direction is the magnetic moment of the particle in the

z-direction.

If we send random particles through the Stern-Gerlach apparatus, which means the

magnetic moments of the particles are randomly distributed, then the force on the

particle should vary continuously from 0 to . So if we have a screen at the exit of the

apparatus, we should observe particles continuously on a range.

The principle of this apparatus is about the same as velocity selectors or

mass spectrometer in physics.

The experiment starts by shooting atoms through this Stern-Gerlach apparatus, as

shown in the graph. The result turns out that only finite impact points are observed on

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the screen, which is different from the prediction of Classical Mechanics. In some special

cases, such as when using silver atoms, there are only two impact points on the screen,

which means it only has two possible magnetic moments. It is because the magnetic

field produced by the outer electrons cancels with the magnetic moment produced by

the angular momentum of the atom. Thus, there must be a property of particle which

produces an extra magnetic field for the silver atom, we call them “spin”.

This simple phenomenon is the one we’ll use to illustrate the magic of Quantum

Mechanics. The particles going to one contact point have the property spin-up. The

other particles (of course, going to another point) have the property spin-down. From

now on we’ll only discuss about particles with two possible spins.

If we denote the Stern-Gerlach apparatus as a box with one entry and two exits (one for

spin-up particles (denoted ) and one for spin-down particles (denoted ), we

could draw the diagram for the experiment described above.

Experiments have shown that if the incoming particles are random, then the particles

exit the apparatus with a half-half chance of being spin-up or spin-down. (Quotation

needed)

The next experiment could be shown as the following:

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In this case, the particles are first split by one Stern-Gerlach apparatus, and then

particles with negative spins are blocked while particles with positive spins are allowed

to enter a second Stern-Gerlach apparatus.

This shows that particles leaving the “+” exit of the Stern-Gerlach apparatus keep its

property when it enter the second Stern-Gerlach apparatus. Thus, all the particles enter

the second Stern-Gerlach apparatus leaves from the “+” exit.

Another experiment shows:

This time the second apparatus is changed into y-direction. It is no surprise that particles

leaving the second apparatus with a half-half chance of being or

Surprises come with the fourth experiment:

In this case, three Stern-Gerlach apparatus are used. Particles exiting the first apparatus

only have property of being . Particles exiting the second apparatus have property

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of being . It may be obvious to think that particles leaving the second apparatus

should have the property of being both and . But these particles turn out to

have property and when they are going through the third apparatus. This

result may lead to the conjecture that particles change their spins every time after the

measurement.

If we consider this experiment in another way, imagine there are many coins floating

freely in the space. If we use two huge, and heavy, plates, one from each extreme of the

z-axis, to project all the coins in the xy-plane, we’ll only find either head or tail, if we

look at the xy-plane from the positive extreme of the z-axis. Now we get rid of the tails,

leaving with all the heads. Then we’ll force these coins onto the yz-plane by apply the

plates from the two extremes of x-axis. We’ll find half of the coins being head, and the

others tail. Now we again apply the force in the z-direction. There is no surprise that half

of them will be head and the other half will be tail, although they used to be all head.

The moral behind this analogue is that measurement (in this case, to see if coins are

head or tail) will change the orientations of the coins.

But the complexity of quantum mechanics is far from coins business above. If we have

forced the coins onto a xy-plane, and suppose all of them are head if we look from the

positive extreme of the z-axis, we apply a force in the direction of the bisector of y-axis

and x-axis. We would expect that all the coins will be heads or tails. But the result of

experiments shows that this coin-analogue can no longer represent the system. If the

coins behave like sub-atomic particles who have spins (in this case, spin-up is head, spin-

down is tail), the result of the experiment shows that there are 14.6% of the coins will

change to tail and 85.4% of the coins will be head. This reveals to us that quantum

mechanics has a probabilistic nature.

Generally, enormous experimental data has showed us that the possibility of finding

head is

, and

for tail, where is the angle between the direction

that the head of coin orients toward before the measurement and the direction we’re

measuring. It is easy to check that if , then we can be 100% sure that all the

heads will remain head and if then we will have a 50%-50% chance of finding

head or tail. We’ll prove this identity at the end of this text.

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The mathematical formulation

Bra-ket notation:

The Bra-ket notation is a standard notation for describing quantum states. A “Bra” is a

horizontal vector. For example, a vector could be notated as . A “ket” is

a vertical vector. For example, a vector could be notated as . Also,

means the inner product (or dot product) between and .

If we have a matrix A, then denotes a vector which comes from implement of

matrix A on the vector b. means the inner product between vector a and the

vector .

Observables and States:

In physics, observables are used to describe a system, more specifically the outcomes of

a system. Observables need to self-adjoint (hemitian matrix). In this context, an

observable could describe the outcome of a Stern-Gerlach apparatus on the incoming

particles. The eigenvalues of an observable tell us the possible values of this

measurement. States are used to describe the property of a system. States are unit

vectors. In this context, a state is sued to describe the property of the particles going

into and leaving a Stern-Gerlach apparatus. The eigenvectors of the associated

observable of a Stern-Gerlach apparatus tell us the states of the particles leaving the

apparatus. The outcome of a measurement system(a) on a system(b) can be calculated

by decomposing the state of the system(b) into linear combination of basis of the

observable of system(a). This will be further explained in the following sections.

Hemitian matrix:

Observables are all hermitian matrix, a hermitian matrix A has the property that .

Also, their eigenvectors are all orthogonal to each other, forming an orthogonal base.

Thus the dot product between one eigenvector and any other eigenvector will give 0.

Pauli Matrix:

The Pauli Matrices are defined as

,

and

. Notice

that the Pauli Matrices are unitary. They could be used as observable in this context.

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Moreover, the eigenvalues of the matrices are 1 and -1, and they have their distinct

eigenvectors.

If there is a Stern-Gerlach apparatus in the x-direction, its action could be described as

the Pauli Matrix,

. If we shot particles with the state

(the eigenvector

of the Pauli Matrix with the associated eigenvalue of 1, denoted as ), we would

get the result vector as 1*

, since

is an eigenvector of that matrix. The square

of the coefficient of this vector tells us the possibility of the particle being in this state

(the state of ). In this case, we call the particles spin-up, because they have the

value +1. Let’s try shot particles with the state

(another eigenvector of ) into

the same Stern-Gerlach apparatus. After the decomposition, we find the result vector to

be

(This is the eigenvector associated with the eigenvalue -1). Similarly, we could

conclude that the possibility that the particle is spin-down is 1 because

What happen if we shot some particles with states which are not eigenvector of the

observable of the system they’re going to? The vector is a eigenvector of the Pauli

matrix , but it is not one of the eigenvector of the Pauli Matrix . To obtain the result

of this operation, we need to decompose the vector into the linear combination of the

eigenvectors of the Pauli Matrix . The result is would be

.

This tells that the particles leaving this apparatus will be with a possibility of 50% of

being

, and a possibility of 50% of being in the state

.

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Spherical polar coordinates:

We can use a spherical polar coordinate system to describe a unit vector in the 3-D

space. We could describe this vector as . This will

facilitate our proof at the end of the text.

Also, with this knowledge, we could compute the observable of a Stern-Gerlach

apparatus oriented in the direction of , the result is

It is easy to check that for special cases like , we will obtain the Pauli Matrices.

The eigenvectors of are:

Where and are eigenvectors of

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Two spins:

Previously we have define to be a spin-up particle in z-direction. We can also

denote a system with two particles as , which means the first particle is spin-up

and the second one spin-down.

If we have a spin-0 particle in the center of a space, and let it split into two particles, we

should expect they have opposite spins in every direction, because of the Conservation

of Angular Momentum.

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Fundamental postulate of quantum mechanics

Postulate I:

The state of a physical system S is completely described by a unit vector , which is

called the state vector, or wave function, and resides in the Hilbert space associated

with the system. The evolution in time of the state vector is governed by the

Schrödinger equation

where H is a self-adjoint operator known as the Hamiltonian of the system and

, with the physical constant h known as Planck’s constant. Its value

( ) is determined experimentally.

This postulate tells us that the state vector of a system changes with time. But it is not

well discussed in this text.

Postulate II:

We associate with any observable A a self-adjoint operator A on the Hilbert space .

The only possible outcome of a measurement of the observable A is one of the

eigenvalues of the operator A. If we write the eigenvalue equation for the operator A,

where is an orthonormal basis of eigenvectors of the operator A, and we expand

the state vector over this basis:

Then the probability that a measurement of the observable A at time t results in

outcome is given by

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This postulate states that the only possible outcome of a measurement is the

eigenvalues of the operator A. In the previous section, we stated that the only possible

outcome of a measurement is the eigenvalue of the Pauli Matrix, which is confirmed by

this postulate. It also points out that a state need to be decomposed in the basis of

eigenvectors of the operator to get the probability that certain outcome take place,

which is what we have done in the previous section.

Postulate III:

If a system is described by the wave vector and we measure an observable A,

obtaining the outcome , then immediately after the measurement the state of the

system is given by

where is the projection operator over the subspace corresponding to .

This postulate tells us that the state will change immediately after the measurement. In

the previous section, we visualize it by saying that the coins change directions every

time after the measurement.

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The Bell’s Inequality

The Bell’s Inequality shows us the incompatibility of two assumptions we make in our

daily life. These two are Reality principle and Locality Principle.

Reality Principle: If we can predict with certainty the value of a physical quantity, then

this value has physical reality, independently of our observation. For example, if a

system’s wave function is an eigenstate of an operator A, namely,

Then the value a of the observable A is an element of physical reality.

This principle states that physical quantities exist independent of our observation. For

example, there are four apples in a basket. Then, regardless of whether we look at it or

not, there are always four apples in a basket. However, as in the previous section, a

positive spin-up in the z-direction doesn’t seem to have physical reality, because it

changes every time we measure it.

Locality principle: If two systems are causally disconnected, the result of any

measurement performed on one system cannot influence the result of a measurement

performed on the second system. Following the theory of relativity, we say that two

measurement events are disconnected if , where and are the

space and time separations of the two events in some inertial reference frame and c is

the speed of light.

If the two measurement are done simultaneously in a very short time (10^-6 seconds)

and in a long distance (1000 km), then any kind of signals of one measurement cannot

reach the other before that measurement is done, if the signal travels at the speed of

light. This implies that the two measurements should be casually disconnected, they

should behave independently. Suppose we have two coins stuck in weightless space,

and they suddenly split into two separate coins, shifting without rotating. One of the

coin fly toward Alice and another toward Bob. After the coins are 1 light-year away, they

will be measured simultaneously by Alice and Bob. They could decide freely in which

direction they’ll measure. Right after Alice’s measurement, the state of the coins (the

state of the coins are just like the state of the two spins, the state contains information

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of the two particles) will change. However, since whichever direction Alice chooses to

measure, the information about Alice’s choice of direction cannot reach Bob. Thus, Bob

will obtains his result whichever direction Alice choose to measure. Thus the two

measurements are casually disconnected.

The Bell’s inequality first admits both of the principle. It starts with a source which emits

particles with opposite spins in every direction. Assume there are three directions,

denoted as a,b,c, in which the particles will be measured by two different persons, Alice

and Bob, far away from each other. If we admit that spins exists independent of our

observation(Reality), then the probabilistic distribution of the state of the particles can

be described by the following table.

Probability Alice's

Particle Bob's

particle

P1 (a+,b+,c+) (a-,b-,c-)

P2 (a+,b+,c-) (a-,b-,c+)

P3 (a+,b-,c+) (a-,b+,c-)

P4 (a+,b-,c-) (a-,b+,c+)

P5 (a-,b+,c+) (a+,b-,c-)

P6 (a-,b+,c-) (a+,b-,c+)

P7 (a-,b-,c+) (a+,b+,c-)

P8 (a-,b-,c-) (a+,b+,c+)

Table1.

We could say that , where all the Ps are positive.

If we denote P(a+,b+) as the possibility that Alice got spin-up in the a direction, and Bob

got spin-up in the b direction, then we could conclude that

We could turn p(a+,b+) into explicit mathematical expression. If Alice finds her particle

spin-up in the a-direction, then the state of bob’s particle could be described by -|a+>,

notice the negative sign. According to Postulate II of the fundamental postulate of

quantum mechanics, we can get that

In this case, means the probability that Bob observes +1 in the b direction. And

also since the probability Alice gets + in the a-direction is 0.5. Thus,

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Also, since Alice’s measurement has collapsed the state of the two-spin, we could

predict with certainty that the state of the two-spin system is , thus the state of

Bob’s particle is .

Thus,

As stated previously, the end of the text will serve to prove the following equation,

We first start with the left-hand side,

,

We could write as the square of the magnitude of inner-product (or dot-

product) between

And

Note that , , Thus, ,

notice the bar above . It is due to the definition of inner product in physics, that we

take the conjugate of our first term.

Thus,

And the product between and can be written as

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Rearrange the terms, we get

Use Euler Identity to expand exponential term, we get

Separate the real part and the imaginary part, we get

Notice that

and

we get

We know that

and

, so we could simplify the expression,

Thus,

Use Half-angle formulas, we could get rid of the fraction in the trigonometry function,

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Simplify it, we get

Use sum and difference formula and simply the expression, we get

Thus,

In spherical coordinate system, , thus

Since a and b are states of a system, they are unit vectors, thus , we get

Finally, we prove that

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If you still remember, we came to an inequality that

for any direction a,b,c.

Now we could rewrite the inequality as

.

And this inequality violates when we chose , where

.

Since there is no error in our calculation, our assumption must be invalid. In this case,

we assume both reality and locality.

Experiments have been done and they have all turned out to have the result which

violate the inequality

Thus, our way of understanding the world, by assuming realism and locality, must be

modified. Either realism or locality should be dropped. Which one would you drop? By

dropping realism, you admit that everything only exists when you observe it. By

dropping locality, you admit that things happening far away are disconnected from

what’s happening in front of you.

Nonetheless, Quantum mechanics precisely explain the behavior of sub-atomic particles,

and thus it could also explain the behavior of big objects, which is composed of

countless of sub-atomic particles, while classical mechanics only gives an approximation

for these behaviors.

Want to understand what Quantum Mechanics tell us about the world around us? Your

journey in realm of Quantum Mechanics has just begun.

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Reference

Jordan, Thomas. Quantum Mechanics in Simple Matrix Form. Dover Publications, Inc.

New York. 1986.

REFERENCE