Download document 8624195

Quantum Physics Lecture 6
Bohr model of hydrogen atom (cont.)
Line spectra formula
Correspondence principle
Quantum Mechanics – formalism
General properties of waves
Expectation values
Free particle wavefunction
1-D Schroedinger Equation
Experimental Evidence for Bohr model
From optical emission spectrum of hydrogen: Consists of line spectra (in contrast to blackbody continuum) Balmer series; fitted to formula
(
1 =R 1 2−1 2
λ
2
n
)
for n = 3, 4, 5....
Experimental value of R = 1.097 x 107 m-1
In general, use two integers ni (initial) and nf (final)
1 =
λ
⎛⎜ 1
⎞
1
R
2 −
2
ni ⎠
⎝ nf
for ni > n f
Where nf = 1 (Lyman), = 2 (Balmer), = 3 (Paschen) etc
Connect expt. with Bohr model
1 = R⎛⎜ 1 2 − 1 2 ⎞
λ
ni ⎠
⎝ nf
4
( )
me
En = − 2 2 1 2
n
8ε o h
Optical emission: result of a “down transition” of electron from a higher energy orbit (ni) to a lower energy orbit (nf)
Energy difference is emitted as a photon:
me4 ⎛ 1
1 ⎞ hc
ω = Eni − En f = − 2 2 ⎜ 2 − 2 ⎟ =
8ε 0 h ⎝ ni n f ⎠ λ
1
me4 ⎛ 1
1⎞
= 2 3 ⎜ 2 − 2⎟
λ 8ε 0 h c ⎝ n f ni ⎠
me4
⇒ R= 2 3
8ε 0 h c
Centre-of-mass correction
me 4
R = 2 3 = 1.097x10 7 m−1
8εo h c
This value of R agreed with expt. of the time! However, later (more accurate) experiments gave Rexpt = 1.0967785
whereas model gave R = 1.0973731
Replace m with m* = mM/(m + M) = 0.99945(m)
In centre-of-mass description of electron (m) and proton (M)
Correspondence Principle
“The greater the quantum number…….
the closer Quantum Physics approaches Classical Physics!”
Correspondence Principle
Compare orbit frequency (f) and emitted photon frequency (ω/2π)
v
e
me 4 ⎛ 2 ⎞
f =
=
=
2 3 ⎜ 3⎟
3
2πr 2π 4 πεo mr
8εo h ⎝ n ⎠
me 4 ⎛ 1
1⎞
ω/2π = E/h = 2 3 ⎜⎜ 2 − 2 ⎟⎟
8εo h ⎝ n f n i ⎠
ν
Note same pre-factor! Write ni = n and nf = n - p
⎛1
⎞ ⎛
⎞ ⎛ 2np − p2 ⎞ ⎛ 2 p ⎞
1
1
1
⎜ 2 − 2⎟ =⎜
⎟=⎜ 2
⎟≈ 3
2 − 2
2
n ⎠ ⎝ n (n − p) ⎠ ⎝ n ⎠
⎝ n f ni ⎠ ⎝ (n − p)
When n >> p, (2np - p2) ~ 2np
and (n - p)2 ~ n2
then ω/2π ~pf and letting p=1, a transition: n to (n-1) gives ω/2π ~ f
Bohr criterion for allowed orbits
The Bohr requirement for orbits can also be stated as:
“angular momentum is quantised in units of ħ”
( 2π ) = n
⇒ 2π r = n ( h ) = nλ
mv
mvr = n h
i.e. Equivalent to “fitting de Broglie wavelengths”
In fact, the concept of quantised angular momentum nħ is the
fuller and broader criterion, with further consequences to be seen;
the other is merely illustrative, not factual!
Complete description of atom requires at least three different
quantum numbers (see later lectures)
General properties of waves
Recall 1-D wave: y = Acos(ωt - kx) This is just one possible solution of the 1-D wave equation:
δ2y 1 δ2y
=
2
v 2 δt 2
δx
In general y = Aexp[−i(ωt − kx )] where i = −1
(ωt − kx
) − isin(ωt − kx )]
= A[cos
- the general (complex) solution.
For ‘waves’ (of existence) in QM use wavefunction ψ
Recall UP and probability: |y|2 is a measure of probability of finding a particle at location x
In QM, ψ is in general complex, and not usually a “measureable” parameter
(such a momentum etc.)
However, |ψ|2 is! So must retain full complex solution, not just real part. General properties of waves cont.
Three other required properties of wavefunction ψ (1) single-valued and continuous
(2) derivative (dψ/dx) single-valued and continuous
(3) normalisable: ∫ |ψ|2dx = 1 - i.e. integrated probability density over all space is unity
i.e. for 1-D case require
If ψ is complex, what about |ψ |2?
∫
+∞
−∞
2
ψ dx = 1
|ψ |2 = ψ*ψ where ψ* is complex conjugate of ψ
ψ = A + iB and ψ* = A - iB ψ*ψ = (A - iB)(A + iB) = A2 - (iB)2 = A2 + B2 (i.e. real)
Expectation values
Expectation value: “the most probable value of a variable”
Multiply variable by probability density (|ψ|2) and integrate! +∞
eg. expectation value of 1-D position:
x
∫
=
∫
x ψ dx
2
−∞
+∞
−∞
ψ dx
If ψ is normalised then denominator = 1, in which case
+∞
x = ∫ x ψ dx
2
−∞
and for a general variable G(x) the expectation value is
+∞
G( x ) = ∫ G( x )ψ dx
−∞
2
2
Free particle wavefunction
….think simple wave, rather than wavegroup……
ψ = Aexp[−i(ωt − kx )]
ν = 2π Eωh = E 
ω = 2π
k = 2π
λ
p
= 2π
h
p
=

⎡ ⎛E
p
⎜
ψ = Aexp⎢−i  t − 
⎣ ⎝
⎞⎟⎤
x⎥
⎠⎦
Replacing “wave-notation” (ω, k) with “particle notation” (E, p)
What is the “equation of motion”, just as in Newton’s Laws, but for quantum particle……?
Free particle functions
⎛ iEt ipx ⎞
ψ = A exp ⎜ −
+
= Ae−iEt  eipx 
⎟
⎝ 
 ⎠
Now consider partial differential with x
∂ψ
ip
−iEt  ip ipx 
= Ae
. e
= ψ
∂x


Re-arranged:
 ∂
ψ = pψ
i ∂x
Or:
∂
−i ψ = pψ
∂x
i.e. if we “operate” on ψ with –iħδ/δx - we get the value of momentum p multiplied by ψ
–iħδ/δx is the momentum ‘operator’
p̂
Expect Kinetic Energy operator
General Equation:
p̂ 2
 2 ∂2
=−
2m
2m ∂x 2
Operator on ψ = value × ψ
1-D Schroedinger Equation
Consider:
Is the equation for energy. ⎡ ⎛
⎤
⎞
p
ψ = Aexp ⎢−i ⎜ E t −
x ⎟⎥
 ⎠⎦
⎣ ⎝ 
∂ψ
= + ip ψ

∂x
2
2
∂ 2ψ
∂
ψ
2
2
p
=−
⇒ p ψ = −
2ψ
2

∂x
∂ x2
∂ψ
∂ψ
E
= −i
ψ
⇒ Eψ = +i

∂t
∂t
Compare with ordinary (non-relativistic) mechanics……
1-D Schroedinger Equation developed
E = KE + PE = p2/2m + U(x,t)
multiply across by ψ
and substitute for Eψ and p2ψ
Eψ
= (p2/2m)ψ + Uψ
∂ψ
 2 ∂ 2ψ
i = −
2 +U
ψ
∂t
2m ∂ x
1-D Schroedinger Equation
∂ψ
 2 ⎛ ∂ 2ψ ∂ 2ψ ∂ 2ψ ⎞
i = − ⎜ 2 + 2 + 2 ⎟ +U ψ 3-D Schroedinger Equation
∂t
2m ⎝ ∂ x
∂y
∂z ⎠
Patently true for free particle (U=0), also found to be true for constrained particle……a basic principle
Steady State Simplification
When U is not a function of t, get considerable simplification:
- The time-independent, or steady state Schroedinger Equation.
Recall free particle wavefunction:
⎡ ⎛
⎞⎤
p
E
ψ = Aexp ⎢−i ⎜
t−
x ⎟⎥
 ⎠⎦
⎣ ⎝ 
⎛ p
⎞
E
ψ = Aexp −i
t exp ⎜ i
x⎟

⎝  ⎠
( )
ψ = ψ′ exp (−i E t )

⎛ p
where ψ′ = Aexp ⎜ i
⎝ 
⎞
x⎟
⎠
Fortunately, this separation of time and position dependences is also
possible for all wavefunctions when U is indep. of t!
…substitute this form of ψ into 1D Schroedinger Equation….
Steady State Schroedinger Equation
2
2
∂ψ
∂
ψ

i
= − 2m 2 + Uψ
∂t
∂x
∂
LHS = i ψ ′ exp −i E  t = Eψ ′ exp −i E  t
∂t
2
2
∂
E t + U ψ ′ exp −i E t
′
RHS = − 
ψ
exp
−i
2m ∂x 2


2
2
∂
ψ′

E
= − 2m exp −i  t
+ Uψ ′ exp −i E  t
2
∂x
2
2
∂
ψ′

⇒ Eψ ′ = −
+ Uψ ′
2m ∂x 2
(
(
(
))
(
(
)
“drop the dash & re-write as”:
(
)
)) (
(
(
))
)
∂ 2ψ 2m
+ 2 ( E −U ) ψ = 0
2
∂x
