Energy Eigenstates
Today’s class:
• Solve Schrodinger equation:
−
• Superposition States
• Get solutions for a bunch of different energies: E1, E2,
E3," (called ‘Energy Eigenstates.’ En:‘Eigenvalues’)
• Different ‘eigenstates’ for each energy:
ψ1(x), ψ2(x), ψ3(x),…
and: Ψn ( x, t ) = ψ n e − iEnt / h
Ψ1(x,t) = ψ1(x)e–iE1t/h,
Ψ2(x,t) = ψ2(x)e–iE2t/h, …
• State with a single energy is called
an “energy eigenstate.”
• ‘Measurements’
Examples of Energy Eigenstates
• Free Particle
ψ(x) = eikx or e-ikx
E = (ħk)2/(2m)
L
From
“Quantum
Bound
States”
simulation
Superposition Principle
• If Ψ1(x,t) and Ψ2(x,t) are both solutions to
Schrodinger equation, so is:
Ψ(x,t) = aΨ1(x,t) + bΨ2(x,t)
• Note: we are still talking about a single electron!
• Examples of superposition states:
– Wave Packet: superposition of many plane waves:
Ψ(x,t) = ΣnAnexp(i(knx-ωnt))
Still only 1 e-!
– Double Slit Interference: superposition of electron going
through left slit and going through right slit:
• Ψtot = Ψ1 + Ψ2
• |Ψtot |2 = |Ψ1 + Ψ2|2 = |Ψ1 |2 + |Ψ2|2 +
Ψ1*Ψ2 + Ψ2*Ψ1
Ψ2
Reminder Superposition
• Infinite Square
Well/Rigid Box
ψn(x) = 2 sin(nπx/L)
From “Quantum Tunneling” simulation
Ψ1
∂Ψ ( x, t )
h 2 ∂ 2 Ψ ( x, t )
+ V ( x ) Ψ ( x , t ) = ih
∂t
2m ∂x 2
Interference Term:
negative → destructive interference
positive → constructive interference
Review of Time Dependence
Say an electron were in the state Ψ ( x, t ) = Ψ1 ( x, t )
where Ψ1(x,t) is the wave function for the ground state of
the infinite square well.
Remember: You can always write an energy eigenstate as
Ψ(x,t) = ψ(x)e–iEt/h.
Probability density = |Ψ(x,t)|2 = Ψ(x,t)Ψ*(x,t)
= ψ(x)e–iEt/hψ*(x)e+iEt/h = ψ(x)ψ*(x) = |ψ(x)|2
⇒ wave function has time dependence in phase.
⇒probability density has no time dependence.
Time dependence of wave function is not
observable. Only probability density is
observable.
Quantum Bound
States sim
Measurement
Time Dependence of Superposition States
An electron is in the state
Ψ ( x, t ) = 12 Ψ1 ( x, t ) + 12 Ψ2 ( x, t ) ,where
Ψ1(x,t) = ψ1(x)e–iE1t/h and Ψ2(x,t) = ψ2(x)e–iE2t/h
are the ground state and first excited
state of
∞
|
Ψ
(
x
,
t
) |2dx = 1 )
the infinite square well. (Note: ∫−∞
• Measurement is a discontinuous process, not
described by the Schrodinger equation.
Does the probability density of the electron
change in time?
• Unlike classical physics, measurement in QM
doesn’t just find something that was already
there – it CHANGES the system!
(Schrodinger describes everything before and after,
but not moment of measurement.)
• If you measure energy of particle, will find it in a
state of definite energy (= energy eigenstate).
• If you measure position of particle, will find it in a
state of definite position (= position eigenstate).
Quantum Bound
States sim
How to compute the probability of
measuring a particular state:
How to compute the probability of
measuring a particular state:
Suppose you have a particle with wave function
Ψ(x,t) = c1 Ψ1(x,t) + c2 Ψ2(x,t) + c3 Ψ3(x,t) + …
Suppose you have a particle with wave function
Ψ(x,t) = c1Ψ1(x,t) + c2Ψ2(x,t) + c3Ψ3(x,t) + …
• Measuring position:
• Measuring position:
b
b
2
2
P(a to b) = ∫ Ψ ( x, t ) dx
P(a to b) = ∫ Ψ ( x, t ) dx
a
a
• Measuring energy:
P(En) = |cn|2
ab
Von Neumann Postulate: If you
• Measuring energy: make measurement of particle in a
state Ψ(x,t), the probability of finding
2
particle in a state Ψa(x,t) is given by:
n
n
ab
P(E ) = |c |
∞
∫Ψ
*
a
( x , t ) Ψ ( x , t )dx
−∞
Note on energy
• Measuring energy " why P(En)
=overlap between
Ψ & Ψa.
Measuring position
= |cn|2
• Example:
double slit experiment:
Ψ(x,t) = c1Ψ1(x,t) + c2Ψ2(x,t) + c3Ψ3(x,t) + …
Ψn(x,t) = cnΨn(x,t)
∞
∫Ψ
*
a
( x , t ) Ψ ( x , t )dx
= ∫ [cn*Ψn*(x,t) Ψ(x,t)]dx
−∞
• Probability density at
screen looks like:
• Probability of measuring
particle at particular pixel:
But sines are orthogonal functions that is:
∞
∫
∫Ψn(x,t) Ψn(x,t) = 0
unless n = m
=
1
if
n=m
*
Ψn ( x, t)Ψ( x, t)dx =
b
−∞
∞
∫cψ
n
−∞
*
n
2
∞
( x)ψ ( x)dx =
∫cψ
n
−∞
|Ψ(x,t)|2
*
n
( x)c nψ n ( x)dx = c n
2
P = ∫ Ψ ( x, t ) dx
a
x
ab
Note on position and energy
measurements:
That’s all on superposition states
and all about the 1D Schrödinger equation.
• Energy eigenstates tend to be spread out in space.
• Position eigenstates tend to be localized in space.
• This is why you can’t know both at the same time (wave
packets vs. plane waves)
• Measuring position messes up energy eigenstate and
vice versa.
Measure Position
Measure Energy
Up next:
Schrödinger in 2D and 3D