Quantum Mechanics for Scientists and Engineers · PDF fileEnergies in quantum mechanics In quantum mechanical calculations we can always use Joules as units of energy but these are

Quantum Mechanics for Scientists and Engineers

David Miller

The particle in a box


Linearity and normalization

Linearity and Schrödinger’s equation

We see that Schrödinger’s equation is linear

The wavefunction appears only in first orderthere are no second or higher order terms

such as 2 or 3

So, if is a solution, so also is athis just corresponds to multiplying both sides by the constant a

2

2

2V E

m

r

Normalization of the wavefunction

Born postulatedthe probability of finding a particle near a point r is

Specifically let us define as a“probability density”

For some very small (infinitesimal) volume d 3r around r

the probability of finding the particle in that volume is

P r 2

r

P r

3P dr r


The sum of all such probabilities should be 1So

Can we choose so that we can use as the probability density

not just proportional to probability density?Unless we have been lucky

our solution to Schrödinger’s equation did not give a so that

3 1P d r r

r 2 r

r 2 3 1d r r


Generally, this integral would give some other real positive number

which we could write aswhere a is some (possibly complex)

numberThat is,

But we know that if is a solution of Schrödinger’s equation

so also is

21/ a

2 32

1da

r r

r

a r


Soif we use the solution instead of

then

and we can use as the probability density, i.e.,

would then be called a“normalized wavefunction”

N a

2 3 1N d r r

2NP r r

2N r

N r


So, to summarize normalizationwe take the solution we have obtained from Schrödinger’s wave equation

we integrate to get a number we call

then we obtain the normalized wavefunction for which

and we can use as the probability density

2 r

21/ a

N a

2 3 1N d r r

2N r

Technical notes on normalization

Note that normalization only sets the magnitude of a

not the phasewe are free to choose any phase for a

or indeed for the original solution a phase factor is just another number by which we can multiply the solution

and still have a solution

exp i


If we think of space as infinitefunctions like , , and

cannot be normalized in this wayTechnically, their squared modulus is

not “Lebesgue integrable”They are not “L2” functions

This difficulty is mathematical, not physical It is caused by over-idealizing the mathematics to get functions that are simple to use

sin kx cos kz exp i k r


There are “work-arounds” for this difficulty1 - only work with finite volumes in actual problems

this is the most common solution2 - use “normalization to a delta function”

introduces another infinity to compensate for the first one

This can be donebut we will try to avoid it


Solving for the particle in a box

Particle in a box

We consider a particle of mass mwith a spatially-varying potential V(z) in the z direction

so we have a Schrödinger equation

where E is the energy of the particleand (z) is the wavefunction

22

22d z

V z z E zm dz

Particle in a box

Suppose the potential energy is a simple “rectangular” potential well

thickness LzPotential energy is constant inside

we choose there rising to infinity at the walls

i.e., at and We will sometimes call this

an infinite or infinitely deep (potential) well

Ener

gy0V

0z zz L

0z zz LzL0V

Particle in a box

Because these potentials at and at are infinitely highbut the particle’s energy E is

presumably finitewe presume there is no possibility of finding the particle outside

i.e., for orso the wavefunction is 0 there

so should be 0 at the walls

Ener

gy

zL

0z zz L

0z zz L

0z zz L

0V

Particle in a box

With these choicesinside the well

the Schrödinger equation

becomes

with the boundary conditions and

Ener

gy

zL0z zz L

0V

22

22d z

V z z E zm dz

22

22d z

E zm dz

0 0 0zL

Particle in a box

The general solution to the equation

is of the form

where A and B are constants

andThe boundary condition

means because

Ener

gy

zL0z zz L

0V

22

22d z

E zm dz

sin cosz A kz B kz

22 /k mE 0 0

0B cos 0 1

Particle in a box

With now and the condition

kz must be a multiple of , i.e.,

where n is an integer

Since, therefore,

the solutions are

with

Ener

gy

sinz A kz 0zL

22 / / zk mE n L

2 2

2kEm

sinn nz

n zz AL

22

2nz

nEm L

1E1n

2E2n

3E3n

Particle in a box

We restrict n to positive integers for the following reasonsSince for any real

number athe wavefunctions with negative nare the same as those with positive n within an arbitrary factor, here -1

the wavefunction for is trivialthe wavefunction is 0 everywhere

Ener

gy

1E

2E

3E

1n

2n

3n

1, 2,n

sin sina a

0n sinn n

z

n zz AL

Particle in a box

We can normalize the wavefunctions

To have this integral equal 1choose

Note An can be complexAll such solutions are arbitrary

within a unit complex factorConventionally, we choose An

real for simplicity in writing

Ener

gy

1E

2E

3E

1n

2n

3n 2 22

0

sin2

zLz

n nz

Ln zA dz AL

2 /n zA L

Particle in a box

Ener

gy

1E

2E

3E

1n

2n

3n

2 sinnz z

n zzL L

22

2nz

nEm L

1, 2,n

zL0


Nature of particle-in-a-box solutions

Eigenvalues and eigenfunctions

Solutionswith a specific set of allowed values

of a parameter (here energy)eigenvalues

and with a particular function associated with each such valueeigenfunctions

can be called eigensolutions

2 sinnz z

n zzL L

22

2nz

nEm L

1, 2,n

Eigenvalues and eigenfunctions

Heresince the parameter is an energy

we can call the eigenvalueseigenenergies

and we can refer to the eigenfunctions as theenergy eigenfunctions

2 sinnz z

n zzL L

22

2nz

nEm L

1, 2,n

Degeneracy

Note in some problems it can be possible to have more than one eigenfunction 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 have definite symmetrythe function is the mirror

image on the left of what it is on the 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 an inverted image the value at any point on the right

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

Such a function has “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 either even or oddi.e., all the solutions have a

“definite parity”Such definite parity is common in

symmetric problemsit is mathematically very helpful

1n

2n

3n

Quantum confinement

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

possible values for the energy2 – there is a minimum possible

energy for the particlecorresponding to

heresometimes called a

“zero-point energy”

1n

2n

3n

Ener

gy

1E

2E

3E

zL0

1n 22

1 / 2 / zE m L

Quantum confinement

3 - the particle is not uniformly distributed over the box, and its distribution is different for different energiesIt is almost never found very

near to the walls of the boxthe probability obeys a

standing wave pattern 1n

2n

3n

Ener

gy

1E

2E

3E

zL0

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 the box where the particle will never be found

1n

2n

3n

Ener

gy

1E

2E

3E

zL0

1n

Quantum confinement

Note that each successively higher energy

state has one more “zero” in the eigenfunction

This is very common behavior in quantum mechanics

1n

2n

3n

Ener

gy

1E

2E

3E

zL0

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 moving through an electrostatic potential change of 1V

Energy in eV = energy in Joules/ee – electronic charge (Coulombs)

191.602 10 J

191.6021 76 565 10 C

Orders of magnitude

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

The first allowed level for the electron is

The separation between the first and second allowed energies ( )

is which is a characteristic size of major

energy separations between levels in an atom

22 10 191 / 2 / 5 10 2.4 10 1.5J eVoE m

2 1 13E E E 4.5eV

Quantum Mechanics for Scientists and Engineers · PDF fileEnergies in quantum mechanics In quantum mechanical calculations we can always use Joules as units of energy but these are

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