School of Mathematical and Physical Sciences PHYS1220 27 August, 20021 PHYS1220 – Quantum...

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School of Mathematical and Physical Sciences PHYS1220School of Mathematical and Physical Sciences PHYS1220

PHYS1220 – Quantum Mechanics

Lecture 4August 27, 2002

Dr J. QuintonOffice: PG 9 ph [email protected]

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The Correspondence PrincipleAny theory must match the well-known laws of classical physics if the conditions match the classical case. This is known as The Correspondence PrincipleRecall from special relativity that when v<<c, the theory must simplify to Newtonian physics eg relativistic kinetic energy becomes

if you take the expansion and make v<<c

In Quantum Mechanics, the same applies in going from microscopic to macroscopic situations (ie when the system >> de Broglie )As the quantum number, n, approaches infinity, any real system should behave in a way that is consistent with classical physics Eg in the Bohr model, the discrete energy levels En get closer and

closer together and as n→, they essentially become ‘continuous’

The same applies for rn and L as n→. The exercise is left to the student

21KE mc 21

2KE mv

1lim 0n nn

E E

2 4

2 2 20

1

8n

Z e mE

h n

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Quantum MechanicsBohr’s model contained a remarkable mixture of classical and quantum concepts, thus it provoked much thought about the wave nature of matter, light and the laws of how they interact with one another

This led to the development of a comprehensive theory to describe microscopic phenomena, started independently in (1925) by Werner Heisenberg (Matrix mechanics, Nobel Prize 1932)

Heisenberg’s approach employs matrices and matrix functions and although very powerful, is mathematically complicated and less suitable for teaching elementary concepts.

Erwin Schrödinger (Wave mechanics, Nobel Prize 1933) Schrödinger’s approach uses multivariable functions and

operators. Their approaches were very different from one another but

their theories are fully compatible

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The Uncertainty PrincipleEvery measurement has an associated uncertaintyAccording to classical physics, there is no limit to the ultimate refinement of the apparatus or measuring procedure It is possible to determine everything to infinite precision

Quantum Theory predicts otherwise. With his matrix mechanics, Heisenberg showed the existence of what is called the Heisenberg Uncertainty PrincipleIf a measurement of position is made with precision x and a simultaneous measurement of momentum component px is made with precision px, then

It is fundamentally impossible to simultaneously measure the exact position and exact momentum of a particle

Other uncertainty relations are

xx p

L E t

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Uncertainty Principle ExampleA ‘thought’ experiment of Heisenberg’s Suppose you want to simultaneously measure the

position and momentum of an electron as precisely as possible with a powerful light microscope

In order to determine the electron’s location (ie making x small ~ ) at least one photon of light (with momentum h/ must be scattered (as in (a))

But the photon imparts an unknown amount of its momentum to the electron (as in (b)), thus altering it’s path and speed! ie p ~ h/ becomes larger!

The very light that allows you to determine the position changes the momentum by some undeterminable amount

In making measurements on microscopic scales, you must now appreciate that you cannot make a measurement without interacting with the very thing that you are attempting to measure!

So x p ~ h

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Is the Bohr Model Realistic?Question: According to the Bohr model, the electron in the ground state moves in a circular orbit with the Bohr radius r1=0.529x10-10m, at a speed of 2.2x106 ms-1 (check it for yourself!). In view of the HUP, is the model realistic?Answer: Because the model assumes that the electron is located at r1, the uncertainty r is zero. If the magnitude of the total momentum of the electron is mv, then the radial component of momentum must be less than or equal to this value

According to the uncertainty principle, the minimum uncertainty in the radial position is therefore

which is ~ r1! so the Bohr model is not realistic

31 6 1 24 1(9.11x10 )(2.2x10 . ) 2.0x10 . .rp mv kg m s kg m s

3410

min 24 1

1.05x10 .0.525x10

2.0x10 . .r

J sr m

p kg m s

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Philosophical ImplicationsNewtonian physics is completely deterministic. If you know the configuration of a system at any point in time then

you can predict its future and also extrapolate its pastQuantum mechanics (QM) has drastically altered our viewpoint Because the wave nature dominates on atomic scales, we must

relinquish determinism and accept a probabilistic approach. The expected position of a microscopic particle (such as an electron

that is moving around a nucleus) can only be predicted by calculating a probability, which in turn indicates an expected statistical average over many measurements.

Even macroscopic objects that are made up of many atoms are governed by probability rather than strict determinism. eg QM predicts a finite (though negligibly small) probability that an

thrown object (comprising many atoms) will suddenly curve upward rather than follow a parabolic trajectory

However when large numbers of objects are present in a statistical situation, deviations from the most probable approach zero, and thus obey classical laws with very high probability, giving rise to an apparent ‘determinism’

Many people opposed this but ultimately had to accept it. At the time Einstein believed that “God does not play dice with the universe”

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School of Mathematical and Physical Sciences PHYS1220School of Mathematical and Physical Sciences PHYS1220The Wave Function and its Interpretation

In Quantum Mechanics, each object (particle or the system itself) is represented by a ‘matter wave’ and is described by a (unique) wave function, (x,y,z,t). The wavelength is established by de Broglie, but what is the physical meaning of the amplitude?One way to interpret the wave function is that it plays the same role that the electric field vector plays in the wave theory of light Recall that Intensity (Amplitude)2

In a similar way, 2 represents an ‘intensity’ or alternatively, the probability of detecting the object with wave function The magnitude of the wave function (itself generally a complex

quantity) may vary in x,y,z or t, but the probability of detecting the particle will be greater where and/or when the amplitude is large.

If represents a single electron (say in an atom) then the value of ||2dV at a certain point in space and time represents the probability of finding the electron within the volume dV about the given position at that time – Max Born, 1928 (Nobel Prize 1954)

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Properties of Wave FunctionsWave functions of particles must possess certain properties to be useful quantum mechanically. The function must be continuousThe function must be differentiablethe particle exists and so the the probability of finding it throughout all of space must be equal to 1. When this is the case, the function is said to be ‘normalised.’ A function must be normalised for the probability to make sense. eg the probability of detecting an electron with a

wave function between x=a and x=b is determined by

Expectation values, < >. The expectation value of any quantity is the statistical average after many measurements of the quantity are made. For example, the expectation value of position <x> is given by

21dV

2b

a

dV

2x x dx

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Eigenvalues and EigenfunctionsConsider

In this example, the derivative is an operator on the function f(x). Because the function f(x) is returned (multiplied by a constant) after it is acted on by the derivative operator, f(x) is said to be an eigenfunction of the derivative operator Or more specifically in this case, the exponential function is an

eigenfunction of the derivative operator

The constant that is returned as a multiplier of an eigenfunction is called its eigenvalue

Here, the constant k is an eigenvalue of the eigenfunction f (x).

Question: is an eigenvector of the 2nd derivative?

( ) kxf x Ae

( ) ( ) ( )kxdg x f x k Ae k f x

dx

( ) kxf x Ae

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Recall from wave theory that any travelling wave can be represented mathematically by

where k is the angular wavenumber, the angular frequency,

A is the amplitude and 0 is an initial phase

Therefore both momentum and energy are contained in the terms describing the wave

Waves Revisited

0( , ) sin( )y x t A kx t

2

2

h hp k

0( , ) sinp E

y x t A x t

22

hE hf f

Now, noticing that

and

then the wave can be expressed by

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The Schrödinger EquationThe Schrödinger equation cannot be derived from first principles. It appears as a postulate, just as Newton’s second law doesWe will consider only one dimensional, steady state problems (where and the potential U are only a function of spatial x and independent of time). The 1-D Time-Independent Schrödinger Equation (T.I.S.E.) is

where m is the mass of the particleThe equation is essentially the total energy of the particle

2 2

2

( )( ) ( ) ( )

2

d xU x x E x

m dx

2 2 2

2 2tot

p kE KE PE U U E

m m

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The Schrödinger Equation II

The T.I.S.E. is based upon conservation of energy, so all 1-D time-independent systems must obey itNote that the T.I.S.E. is an operator equation, however.The wave function of any real object must be an eigenfunction of Schrödinger’s equation, with its corresponding eigenvalue equal to the object’s energy, E. In other words, once you know the eigenfunction of a

particle (or its state) you can just substitute it into Schrödinger’s equation to calculate the energy

Well that is quite a bit of theory, now let’s use it for some simple situations

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Free ParticlesThe simplest wave function to describe is that of a free particle. It does not have any potential acting on it and therefore no forces.The T.I.S.E. is therefore

Which can be written

This is the second order differential equation for a harmonic oscillator with general solution

Note that k can have any value (ie the energy can be chosen from a continuous range). Note also that px is zero, so x →

2 2

2

( )( )

2

d xE x

m dx

2

2 2

( ) 20

d x mE

dx

2( ) sin cos ,

mEx A kx B kx k

2 2 2

21

2 2 2

p kE mv

m m

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Free Particles IIGiven the wave function, let’s substitute it back into the T.I.S.E.

So

If we had guessed the wave function, we could have computed the energy of the free particle (and how it depends upon k)That is a simple example of the power of Schrödinger’s equation

( ) sin cosx A kx B kx

2 2 2

22

22

( )sin cos

2 2

( ) ( )2

d xk A kx B kx

m dx m

k x E xm

22 2,or alternatively

2

mEE k k

m

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Wave PacketsTo represent a particle that is well localised (ie its position is known to be within a small region of space), we use the concept of a wave-packet

To describe this requires a wave function that is the sum of many sinusoidal plane waves of slightly different wavelengths (cf beats). The smaller the value of x, the more terms are

needed in the sum.

Because each term in the sum has a unique wavelength (and therefore momentum), the sum does not have a definite momentum. Rather, it has a range of momenta, so px is non-zero

School of Mathematical and Physical Sciences PHYS1220 27 August, 20021 PHYS1220 – Quantum...

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