Download - Ch5 Matter waves - ocw.snu.ac.kr
Matter waves• We were able to derive the wave equations for waves in a
string or for electromagnetic waves from a knowledge of the medium in which the waves propagate.
• We construct a formal equation that will meet very general requirements that matter often exhibits wave-like properties, such as electron diffraction.
• How can we construct the wave equation for matter waves?
Matter waves• The desired solution may have the form like
• The features the matter wave equation should include are the energy Eand the momentum p, which are common between the “particle” and the “wave”.
• If we assume that the matter wave equation has the harmonic solutions as the above, then
( ) (1)i kx tAe
𝐸 𝑝
particle 𝑝2𝑚 𝑉 𝑚𝑣
wave ℏ𝜔 ℏk
2 2 22 2
2 2
(2)
(3)
ii Et t
k px x
Schrödinger time dependent wave equation
• Since the energy of a particle is given by
• The above equation is the Schrödinger time-dependent wave equation.
• The equation must be used if we want to describe processes related to dynamic transition, such as in optical absorption and carrier relaxation.
2
2 2
2
22
( 2 ) ,a reasonable choice of a wave equation is
1 2
. . (4)2
E p m V
i Vt m x
i e i Vt m
Schrödinger time-independent wave equation
• If we are not concerned with the dynamic changes of energy state, but concerned with the question of what energy states are allowed in the presence of a particular potential energy V(x, y, z), so called stationary states of the system, the states can be expressed as
• Substitution of the equation (5) into (4) leads to
Ψ 𝑥, 𝑦, 𝑧, 𝑡 𝜓 𝑥, 𝑦, 𝑧 𝑒 5𝑖. 𝑒. The dependence of Ψ on the coordinates can be separated from the dependence of Ψ on time.
ℏ𝜔𝜓 𝐸𝜓ℏ2𝑚
𝑑 𝜓𝑑𝑥 𝑉 𝑥 𝜓 for 1−D
or ℏ2𝑚
𝑑 𝜓 𝑥𝑑𝑥 𝑉 𝑥 E 𝜓 𝑥 0 for 1−D 6
or ℏ2𝑚 ∇ 𝜓 𝑉 𝑥, 𝑦, 𝑧 E 𝜓 0 for 3−D 7
Time−independent Schrodinger equation
Schrödinger time-independent wave equation
• Procedure for solving the wave equation
1. Obtain the general solution 𝜓 𝑥 for the particular V of interest.
2. Retain only mathematically well behaved solution, i.e.
3. Apply boundary conditions.
. . obtaining a set of with corresponding allowed energy values i ii e E
single valuednot zerocontinouscontinous derivativesfinitesquare integrable
may limit nE
may limit nE
• What is 𝑥 ?• Physical significance of 𝑥 can be associated at least with
the real quantity .• Suppose that a plot of vs x representing a
“particle” has the dependence on x shown in the below figure.
2*
2
x0x
*
• Two interpretations of① The probability of finding the particle between x and x+dx
or within dxdydz of (x, y, z).<“Where is the particle likely to be”>
② Represent the density function for the “particle” e.g., represents a spatial distribution of charge
corresponding to a single electron. collapse to a specific location during measurement with
a probability ∝ density function The probability of measuring a particular point is
proportional to the magnitude of the density function at that point.
<Particle does not have a position>
2 .dxdydz
2q
2
• Since in either interpretation, is a probability, it must be
“square integrable”.
• When the wave function is multiplied by an appropriate constant A such that
• The wave function is said to be normalized, and A is called the normalization constant.
• A normalized wave function satisfies
2
𝐴 𝜓 𝑑𝑥𝑑𝑦𝑑𝑧 1
𝜓 𝑑𝑥𝑑𝑦𝑑𝑧 constant
𝜓 𝑑𝑥𝑑𝑦𝑑𝑧 1 8
• To be used as a probability, the wave function must be normalized.
• In spherical coordinates
• Three important systems allow exact solutions.
① A free electron model of a confined electron
② Linear harmonic oscillator (F=-gx)
③
𝑒. 𝑔. 𝑞 𝜓 𝑑𝑥 𝑞 9
𝜓 𝑟, 𝜃, 𝜙 𝑟 sin 𝜃 𝑑𝑟𝑑𝜃𝑑𝜙 1 10
𝑉12 𝑔𝑥 ,
𝑉𝑍𝑞
4𝜋𝜀 𝜀 𝑟 , Hydrogenic atom 𝑍 1, hydrogen
0 for 0 ,V x L
Free electron model of a confined electron
• The Schrödinger equation becomes
• The solution of the above equation has the form.
• Substitution eq. 12 into eq. 11 results in
2 2
2
( ) E ( ) 0 (11)2
d x xm dx
( ) (12)ikx ikxx Ae Be
2 2
, 02
kE x Lm
0 L
0V V V
for 0, 0
0 for 0outside
V x x L
x L
1-D case“Particle in a box”
x
Free electron model of a confined electron
• B.C’s①
②
, 2
sinn n
n nk LL
n xCL
1/2*
0
1/2
22 2 2 2
2
21
2 sin
2 2
L
n
n
n
dx CL
n xL L
n nEm L mL
0 at 0 and x L
Free electron model of a confined electron
Number of nodes in More nodes correspond to the higher energy.
2
1/22 2
2
1/22 2
2
21 sin2
ground state
2 22 4 sin2
n n n nn E
xmL L L
xmL L
1/22 2
2
2 33 9 sin2
L
xmL L L
( 1)n
Free electron model of a confined electron
For large n2
n
Classical① Equal probability of
finding particle anywhere (for any E) = 1/L
② Continuous E 0 allowed
WaveProbability is function of x and varies with E
Discrete En allowed
③ As wave picture reduces to classical picture.“correspondence principle”
,n
15 21 3.8 10 eVE L (L in centimeters)
Linear harmonic oscillator
- solution :
𝐹 𝑔𝑥
𝑉 𝐹𝑑𝑥12 𝑔𝑥
ℏ2𝑚
𝑑 𝜓𝑑𝑥
12 𝑔𝑥 𝐸 𝜓 0
𝜓 𝑥 𝑓 𝑥 𝑒 when 𝛾𝑚𝑔ℏ
𝑓 𝑥 : p olynomial that terminate after a finite number of terms to keep 𝜓 𝑥 finite and normalizable
𝜓 𝐻 𝛾𝑥 𝑒
𝐻 𝑦 1 𝑒𝑑
𝑑𝑦 𝑒
:Consists of a particle moving under a restoring force proportional to the displacement.
From termination condition :
• Applications
𝐸 𝑛12 ℏ𝜔 𝜔 2𝜋
𝑔𝑚
phononphonon
k
Linear harmonic oscillator
① Vibration of atoms in a crystal can be represented by a collection of oscillators representing lattice waves. →
② Collection of charge oscillators generates electromagneticradiation →
Linear harmonic oscillator
photonphoton
𝑛 𝜓 𝐸 𝜓 𝜓
0 𝐴 𝑒 12 ℏ𝜔
1 𝐴 𝛾 𝑥𝑒 32 ℏ𝜔
2 𝐴 2𝛾𝑥 1 𝑒 52 ℏ𝜔
• Classically, mid point has lowest probability, side point has largest probability.→ not the case in quantized oscillator.
-For large n
Linear harmonic oscillator
𝜓
𝜓Max. probability at the classical “turn around” points, at the “ends” of the oscillation.
Constant probability
[Particle in a box]
[Linear harmonic oscillator]