nature of particle-in-the-box solutions

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2.3 Particle in a box

Slides: Video 2.3.6 Nature of the

particle-in-a-box solutionsText reference: Quantum Mechanics

for Scientists and EngineersSection 2.6 (second part)



The particle in a box

Nature of particle-in-a-box sol

Quantum mechanics for scientists and engineers Dav



Eigenvalues and eigenfunctions

Solutionswith a specific set of allowed values

of a parameter (here energy)eigenvalues

and with a particular functionassociated with each such value

eigenfunctionscan be called eigensolutions

2

n

z

z L

2

2n E

m

1, 2,n



Eigenvalues and eigenfunctions

Heresince the parameter is an energy

we can call the eigenvalueseigenenergies

and we can refer to theeigenfunctions as the

energy eigenfunctions

2

n

z

z L

2

2n E

m

1, 2,n



Degeneracy

Note in some problemsit can be possible to have more than oneeigenfunction with a given eigenvalue

a phenomenon known as

“degeneracy”The number of such states with the same

eigenvalue is called“the degeneracy”of that state



Parity of wavefunctions

Note these eigenfunctions havedefinite symmetrythe function is the mirror

image on the left of what it is onthe rightsuch a function has “even parity”

or is said to be an “even function”The eigenfunction is also even 1n

2n

3n 1n

3n




The eigenfunction is aninverted imagethe value at any point on the right

of the centeris exactly minus the value at the“mirror image” point on the left

of the centerSuch a functionhas “odd parity”or is said to be an “odd function”

1n

2n

3n

2n




For this symmetric well problemthe functions alternate between

being even and oddand all the solutions are eithereven or oddi.e., all the solutions have a

“definite parity”Such definite parity is common insymmetric problemsit is mathematically very helpful

1n

2n

3n



Quantum confinement

This particle-in-a-box behavior isvery different from the classical case1 – there is only a discrete set of

possible values for the energy

2 – there is a minimum possibleenergy for the particle

corresponding toheresometimes called a

“zero-point energy”

1n

2n

3n

E n e r g y

01n 22

1 / 2 / z E m L



Quantum confinement

3 - the particle is not uniformlydistributed over the box, andits distribution is different fordifferent energies

It is almost never found verynear to the walls of the box

the probability obeys astanding wave pattern 1n

2n

3n

E n e r g y

0



Quantum confinement

In the lowest state ( ),it is most likely to be found

near the center of the boxIn higher states,

there are points inside thebox where the particle will

never be found 1n

2n

3n

E n e r g y

0

1n



Quantum confinement

Note thateach successively higher energy

statehas one more “zero” in theeigenfunction

This is very common behavior in

quantum mechanics 1n

2n

3n

E n e r g y

0



Energies in quantum mechanics

In quantum mechanical calculationswe can always use Joules as units of energy

but these are rather largeA very convenient energy unit

which also has a simple physical significanceis the electron-volt (eV)

the energy change of an electron in movinthrough an electrostatic potential changeEnergy in eV = energy in Joules/e

e – electronic charge (C

191.602 10 J

191.602176 565 10 C



Orders of magnitude

E.g., confine an electron in a 5 Å (0.5 nm) thbox

The first allowed level for the electron is

The separation between the first andsecond allowed energies ( )is

which is a characteristic size of majoenergy separations between levels i

an atom

22 10 19

1 / 2 / 5 10 2.4 10 1.5Jo E m

2 1 13 E E E

4.5eV

nature of particle-in-the-box solutions

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