Postulates of Quantum Mechanics | PHY 571, Study notes of Quantum Physics

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John Venables

5-1675, [email protected]

Spring 2008

The Transition to Quantum Mechanics

Module 1, Lecture 6, 7 and 8

PHY 571: Quantum Physics

Postulates of Quantum Mechanics

•^

Describe waves/particles via the Wave function,

ψ(

r ,t

-^

Wave function evolves in time (Schrödinger) or the Operatorevolves in time (Heisenberg): equivalent, but they appear to bevery different (

several math tools, interacting with one physics;

many to one relationship, discuss...

)

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Interpretation of the wave function (Born): Probability, P, ofobserving particle is proportional to |

ψ

(^2) |; normalized to P = 1

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Schrödinger equation (SE) is a (deterministic) operatorequation; the order of the operations is important: it "operates"on the wave function. Basic is the time-dependent SE, but mostuseful (in materials/ nanoscience) is the time-independent SE.

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Download Transition to Quantum Mechanics document

The transition to Quantum Mechanics

☺

We want to make the transition from particles

r

(t) and

waves

ω

( k

) to the wave function

ψ(

r ,t

)^

which is

either/or (or is it) both/and...

a)

This function

ψ(

r ,t

)^

can be associated with (a spread of)

energies E =

=

ω

and momenta

p

=

=

k

b)

ψ(

x,t

)^

satisfies: i

=∂/∂

t(

ψ) = −

(=

2 /2m)

(^2) ∂ ψ/∂

(^2) x

• V(x)

ψ:

this is the time-dependent SE in 1D

c)

Probability P(

r ,t) = |

ψ

(^2) |

=^

ψ

∗^ ψ

(of finding particle at

position

r

and time t)

/

Final element is the role of Measurement which"collapses

ψ(

r ,t

)^

onto an Eigenstate"...

not trivial, take

your time to understand all this

Do the various authors agree on pedagogy?

•^

Feynman lecture video: guess

prescription (model);

prescription

consequences, comparison with expt.

•^

Griffiths book

states

b) and c) on p1-2; Gasiorowicz

argues towards

b) and c), but only for a free particle

(chapter 2).

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So either way it is a guess, a jump, a

Quantum Leap

( Life looks different after you have taken the bait

): a

fishy story, but now you see Classical Mechanics as the incoherent

limit of Quantum Mechanics.

•^

In QM we add amplitudes and then take

ψ

∗ψ

to get

(probabilistic) intensities; in CM we add intensities;understand the importance of phases (coherence)

Example: The Importance of Phases

•^

The 2-slit interference experiment:

If

ψ

1

and

ψ

2

are

coherent, we add amplitudes and then form intensities.

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Exercise: take

ψ

1

= Aexp(ik

y) and 1

ψ

2

= Bexp(ik

y), 2

where A and B can be complex (i.e. can have phases

φ

1

and

φ

2

respectively).

•^

Work out the intensity I =

ψ

∗ψ

, where

ψ = ψ

1

• ψ

keeping proper track of all the complex numbers

-^

Express the answer in terms of trigonometric functions (cos, sin,etc) if it simplifies, and if it doesn't, find the conditions underwhich it does (e.g. |A|=|B|, A=B, or A = B = 1).

Note I is real...

-^

Keep your (right) answers at your bedside for future reference..

Expectation values, Averages and Operators•^

Probability P =

ψ

∗^ ψ: Β

ut how is this related to classical

probability? In classical physics the order of operationsdoesn't matter, but in quantum physics it does...

-^

(t) the Average =

∫f(x)P(x,t)dx in classical physics.

In quantum physics the same quantity is the

-^

Expectation Value =

∫ψ

∗ f(x)

ψ

dx;

note order

•^

f(x)

operates

on

ψ.

Operators may

not commute

-^

Examples: the free particle SE, operators for

p

, x or

r

V(

r ) and E; Relationship to measurement and the Uncertainty Principle; Poisson brackets andRepresentations (x and

p

). ...

Whoa again!

Example: back to the Current operator, j(

r

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Probability current and particle number conservation leads to

P

t +

j (

r ) = 0, that works for free particles

and when V(

r ) is real

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1D result:

j

( x

/2im)(

ψ

∗^

x(

ψ) − ψ∂/∂

x(

ψ

∗)

Exercise:

Write down the 3D result by analogy, and

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Try this out on a beam in free space, where

ψ(

r ,t

ψ

exp[i( 0

k

. r

• ω

t)] and show that

j

k

/m)

ψ

∗ 0 ψ

. This 0

for real V(

r ), vectors in 3D; start with 1D:

r

x,

k

k

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If there is an imaginary part of the potential V

(i r ), there

is absorption,

k

k

+i

q

, and then (in 1D) the current

j(x) reduces with distance: j = (

k/m)

ψ

∗ 0 ψ

exp( 0

2qx)

Relationship of operators to H's UP

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If operators commute, can be measured simultaneouslyExample p and H. Check that [pH-Hp]

ψ

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If operators do not commute,

cannot be measured

simultaneously

, the quantities obey H's UP. Example x

and p

. Check that [xpx

-px

x]x

ψ

/i)

ψ

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[ ] is called a commutator, and is written [p,H] or[x,p

]. The commutator [x,px

] =x

/i states Heisenberg's

Uncertainty Principle precisely: an

Operator equation

•^

In classical physics [ ] is called a Poisson bracket. As =

0 we get classical results. One way to start quantum

mechanics.

Operator theorems

, true for any

ψ