6.1 The Schrödinger Wave Equation
6.2 Expectation Values
6.3 Infinite Square-Well Potential
6.4 Finite Square-Well Potential
6.5 Three-Dimensional Infinite-Potential Well
6.6 Simple Harmonic Oscillator
6.7 Barriers and Tunneling
CHAPTER 6
Quantum Mechanics II
I think it is safe to say that no one understands quantum mechanics.
Do not keep saying to yourself, if you can possibly avoid it, “But how can it be like that?” because you will get “down the drain” into a blind alley from which nobody has yet escaped. Nobody knows how it can be like that.
- Richard Feynman
Origin of Quantum Mechanics is credited to Werner Heisenberg and Erwin Schrödinger.
Heisenberg Matrix formalism (1925).
Schrödinger Wave Mechanics (1926).
similar to wave descriptions in classical physics.
Both methods give identical results.
QM complex subject and probabilistic nature is in contrast to cause and effect of classical physics.
In this chapter will use QM for some “simple cases”.
Quantum Mechanics
Thomson – Plum Pudding
Why? Known that negative charges can be removed from atom.
Problem: just a random guess
Rutherford – Solar System
Why? Scattering showed hard core.
Problem: electrons should spiral into nucleus in ~10-11 sec.
Bohr – fixed energy levels
Why? Explains spectral lines.
Problem: No reason for fixed energy levels
de Broglie – electron standing waves
Why? Explains fixed energy levels
Problem: still only works for Hydrogen.
Schrödinger – probability distribution
Models of the Atom
– – – –
–
+
+
+ –
6.1: The Schrödinger Wave Equation
If particles exhibit wave properties (de Broglie) then, like any other wave phenomena, there should be a wave equation for a particle (a differential equation which describes the wave propagation).
Like Newton’s laws in classical physics for particles we need an equation to describe their wave motion.
Schrödinger found an equation for the wave function using ideas from optics.
the solution to the wave equation (the wave function) must be consistent with the notion of the uncertainty principle and the notion of probability.
Like Newton’s laws in classical physics there is no way to derive the Schrödinger wave equation from more basic principles.
the ultimate test is to see if it describes experimental results.
Schrodinger equation development new approach to physics
Wave Equations
EM Waves (light/photons)
• Amplitude = E or M field
• ||2 tells you the probability of detecting a photon.
• Maxwell’s Equations:
• Solutions are sine/cosine waves:
2 2
2 2 2
1
x c t
( , )x t Asin(kx t)
( , )x t Acos(kx t)
Matter Waves (electrons/etc)
• Amplitude = matter field
• ||2 tells you the probability of detecting a particle.
• Schrödinger Equation:
• Solutions are complex sine/cosine waves:
i t x
m
2 2 2
2
( , )x t Aexp i kx t
cos( ) sin( )
A kx t i kx t
differential equations which describe the propagation of waves
Schrodinger’s starting point:
2 2
2 2 2
1
x c t
2
2
E p V
m
Which aspects of a particle wave equation need to be similar and which different from classical wave equations?
Not going to derive it, because there is no derivation…
Schrodinger started to think………
Works for light, why not for an electron?
2 2 2
E p c 2 2 2
2 c 2
t x
: ~ partial derivative with respect to : ~ partial derivative with respect to
E E t
t
p p x
x
For electron,
2 2
So, one may guess that
~ 1
2 V
t m x
Schrodinger’s starting point:
The Planck–Einstein and De Broglie relations: E
, p k
( , )x t Aexp i kx t Aexp i px Et /
( , )
( , ) x t i
p x t i p p i
x x x
( , )
( , )
x t i
E x t i E E i
t t t
2 2 2
2
( , ) ( , )
( , ) ( , )
2 2
p x t x t
E V i V x t x t
m t m x
: ~ first partial derivative with respect to : ~ first partial derivative with respect to
E E t
t
p p x
x
Complex wave solution:
There is no way to derive the Schrödinger wave equation from more basic principles.
the ultimate test is to see if it describes experimental results.
The Schrödinger Wave Equation
The Schrödinger wave equation in its time-dependent form for a particle of energy E moving in a potential V in one dimension is
The extension into three dimensions is
Schrödinger Eqn. is linear in the wave function Ψ (example 6.1).
this required since principle of superposition is used to form wave packets
General Solution of the Schrödinger Wave Equation
The general form of the solution of the Schrödinger wave equation is given by:
which also describes a wave moving in the +x direction.
In general the amplitude may also be complex.
This is called the wave function of the particle.
The wave function must not be restricted to being real.
Notice that the sine term has an imaginary number.
Only the physically measurable quantities however must be real.
Can A exp(kx + wt) be also a solution to the equation?
Normalization and Probability
The probability P(x) dx of a particle being between x and X + dx was given in the equation
here denotes the complex conjugate of
The probability of the particle being between x1 and x2 is given by
The wave function must also be normalized so that the
probability of the particle being somewhere on the x axis is 1.
*
Example 6.4
Properties of Valid Wave Functions
Boundary conditions
1) In order to avoid infinite probabilities, the wave function must be finite everywhere.
2) In order to avoid multiple values of the probability, the wave function must be single valued.
3) For finite potentials, the wave function and its derivative must be continuous.
This is required because the second-order derivative term in the wave equation must be single valued. (There are exceptions to this rule when V is infinite.)
4) In order to normalize the wave functions, they must approach zero as x approaches positive and negative infinity.
Solutions that do not satisfy these properties do not generally correspond to physically realizable circumstances.
( ) ( )
at boundary
( ) ( )
I II
I II
x x
x x
x x
Time-Independent Schrödinger Wave Equation
The potential in many cases will not depend explicitly on time.
The dependence on time and position can then be separated in the Schrödinger wave equation.
Let ,
which yields:
Now divide by the wave function:
The left side of this last equation depends only on time, and the right side depends only on spatial coordinates. Hence each side must be equal to a constant. The time dependent side is
We integrate both sides and find:
where C is an integration constant that we may choose to be 0.
Comparing with wavefunction for a free particle, A ei(kx –t):
Plugging back into spatial equation yields:
This is known as the time-independent Schrödinger wave equation,
It is a fundamental equation in quantum mechanics.
The wave function (with time dependence) can be written:
Time-Independent Schrödinger Wave Equation (con’t)
B E
( , ) x t ( ) x e
i t
Stationary State
Recalling the separation of variables:
and with f(t) = the wave function can be written as:
The probability density becomes:
The probability distributions are constant in time.
This is a standing wave phenomena that is called the stationary state.
(x,t)
(x) f (t)e
itComparison of Classical and Quantum Mechanics
Newton’s second law and Schrödinger’s wave equation are both differential equations.
Both postulated as fundamental equations
to explain observed behavior and verified by experiment.
Newton’s second law can be derived from the Schrödinger wave equation, so the latter is the more fundamental.
Classical mechanics only appears to be more precise because it deals with precise values rather than probabilities.
• only appears precise because it deals with macroscopic phenomena.
The underlying uncertainties in macroscopic measurements are just too small to be significant.
• Classical mechanics is accurate enough at large quantum numbers, but as far as we know, there is only one correct theory, QM.
