PYL 100 2016 QMLect 05 Ststate InfSqwell

8/17/2019 PYL 100 2016 QMLect 05 Ststate InfSqwell

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PYL100 Electromagnetic Waves and Quantum Mechanics

STATIONARY STATES

The general solution is

Take that to

Time-independent Schrödinger equation

We can go no further with it until the

potential V (x) is specified.

What's so great about separable solutions?

After all, most solutions to the (time dependent) Schrodinger

equation do not take the form (x)(t).


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STATIONARY STATES

Probability density: ,

does not depends on time

A stationary state

Same thing happens in calculating the expectation value of any dynamical

variable:

Every expectation value is

constant in time

= Constant 0


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States of definite total energyIn classical mechanics, the total energy

(kinetic plus potential) is called theHamiltonian:

In quantum mechanics,

Hamiltonian operator:

Time-independent Schrödinger equation

Expectation value of the total energy is


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States of definite total energy

Expectation value of the total energy is

Conclusion: A separable solution ( (x) (t)) has the property that

every measurement of the total energy is certain to return the value E.

(x) is a state of definite total energy:E(x)


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Summary

Given a (time-independent) potential V(x), and the starting wave

function (x, 0); how to find the wave function, (x, t)

One need to solve

The first strategy is to solve

This yields, in general, an infinite set of solutions ( 1(x), 2(x), …)

each with energy (E1, E2, …)

Then find (x, 0);

Construct (x, t);


7/20PYL100 Electromagnetic Waves and Quantum Mechanics

Example


8/20PYL100 Electromagnetic Waves and Quantum Mechanics

Help session

Wednesday, 5 p.m. MS-420, & MS-418


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THE INFINITE

SQUARE WELLChapter 2.2

Introduction to QuantumMechanics (2nd Edition)

David J. Griffiths


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THE INFINITE SQUARE WELLA particle is completely free

Not free at x = 0 and x = a

• A classical Analog: A cart on a frictionless horizontal air track, with perfectly

elastic bumpers-it just keeps bouncing back and forth forever.

•

This potential is artificial-but treat it with respect.


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THE INFINITE SQUARE WELL

Outside: the probability of

finding the particle is

zero=> =0

Inside the well, where V = 0


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zero=> =0



Arbitrary constants

Typically, these constants are fixed by the boundary conditions of the problem.

What are the appropriate boundary conditions for (x)?


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zero=> =0



Arbitrary constants

Typically, these constants are fixed by the boundary conditions of the problem.

What are the appropriate boundary conditions for (x)?

Both (x) and d (x) /dx are continuous expect for = ∞


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Arbitrary constants

Both (x) and d (x) /dx are continuous expect for = ∞

Continuity of (x) requires that: (0) = (a) =0

==> (0) = B=0 ==> =

==> = = 0

A = 0 => = 0

trivial-non-

normalizable-

solution,

= 0

=> X

The negative solutions give nothing new,

since sin − = −sin() and we can absorb

the minus sign into A.


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Energy Quantization

So the distinct solutions are

Hence the possible values of E:

A quantum particle in the infinite square well cannot have just any

energy-it has to be one of these special allowed values: In contrast to the

classical case.

To find A, we normalize :


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Zero-point energy

is called the zero-point energy .

• It is the lowest possible total energy the particle bound by the

infinite square well potential.

• The particle cannot have zero total energy.

• The phenomenon is basically a result of the uncertainty principle

• Uncertainty in position is a

• For the particular case of eigenvalue E 1 , the magnitude of the

momentum is 1 = 21 = ±ℏ

It can be moving in either direction.

The actual value of the momentum is uncertain by an amountwhich is about ∆ ≈ 21= 2

ℏ

=> ∆∆ ≅

ℏ

= 2ℏ

agrees with uncertainty principle

The energy of the first eigenvalue

Application example: Helium does not solidify even at the lowest

attainable temperature (1 mK), unless a very high pressure is applied.


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Wave functions

Infinite set of solutions (one for each positive integer n).

They look just like the standing waves on a string of length a

Lowest energy:ground state

First Excitedstate

Second Excitedstate


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Important Properties of

1. They are alternately even and odd, with respect to the

center of the well: is even, is odd, is

even, and so on

2. As you go up in energy, each successive state has one more

node (zero-crossing): has none (the end points don't

count), has one, has two, and so on.

3. They are mutually orthogonal:

whenever ≠


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Proof: Mutually orthogonal

In general,


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Important Properties of

4. They are complete, in the sense that any other

function, f (x), can be expressed as a linear combination

of them:

From mathematics: Fourier series for f (x): Dirichlet's theorem

What is the coefficients cn ?

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PYL 100 2016 QMLect 05 Ststate InfSqwell

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