1
Classical Mechanics
Newton’s Second Law tells us how to find information about the future state of a
Classical Mechanical system from its present state:
2
2
dt
xd
mmaF
2
2
dt
xd
dt
dx
dt
d
dt
dv
a
Acceleration:
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Understanding Quantum Mechanics: From Classical Mechanics to Schrödinger Equation - Prof. , Study notes of Physical Chemistry
An overview of the development of quantum mechanics, starting from newton's second law and classical mechanics. It covers the wave-like nature of matter, the classical nondispersive wave equation, and the schrödinger equation. The text also introduces the concept of operators and eigenvalues in quantum mechanics.
Typology: Study notes
2010/2011
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Classical Mechanics Newton’s Second Law tells us how to find information about the future state of a Classical Mechanical system from its present state: 2 2
dt
d x
F ma m
2 2 dt d x dt dx dt d dt dv a (^) Acceleration: ^
2 Development of Quantum Mechanics Starting Point: Classical Wave Mechanics Wave-like nature of matter suggests that a wave equation should be used to describe atoms, molecules, and electrons. wave vector 2 k angular frequency^ ^ ^2 ^ phase shift In general, for a wave propagating in a non-dispersive medium (at velocity v): Classical Nondispersive Wave Equation
Development of Quantum Mechanics Relevant Mathematical Concepts New: Operators Eigenvalues, Eigenfunctions, Wavefunctions Review: Complex numbers Trigonometry & Euler relation Calculus (partial differential equations, integration)
Quantum Mechanics Strategy to understanding QM: Postulate the basic principles and use these postulates to deduce experimentally testable consequences. e.g. Energy levels of atoms To describe the state of a system in Quantum Mechanics, postulate the existence of a function of the particles coordinates called the wavefunction ( Y ). Y Y x , t For a one dimensional system:
Y contains all of the possible information about the system.
The Postulates of Quantum Mechanics Postulate 2: Every Observable has a Corresponding Operator Quantum mechanics can be expressed in terms of six postulates which summarize the principle tenets of quantum mechanics. Postulate 1: The Physical Meaning of the Wavefunction Postulate 3: The Result of any Individual Measurement Postulate 4: The Expectation Value Postulate 5: The Time Evolution of a Quantum Mechanical System Postulate 6: The Symmetry of the Wavefunction WRT Parity
Postulate 1: The Physical Meaning of the Wavefunction That is, the wavefunction contains all of the dynamical information about the system it describes. The state of a quantum mechanical system is completely specified by a
wavefunction Y (x,t).
The Born Interpretation of the Wavefunction: Focus on Location: At any given point in time, what is the position of the particle? The probability of finding the particle at time t 0 in a region of space of width dx centered at x 0 is given by: P x t x t x t dx x t dx 2 0 0 0 0 0 0
0 0 , Y , Y , Y ,
Postulate 1: The Physical Meaning of the Wavefunction
The Properties of the Wavefunction ( Y or )
i. must be normalized.
In order for the wavefunction to be a plausible solution to the Schrödinger of real system (i.e. physical), the following restrictions hold:
ii. must be finite.
iii. must be continuous.
iv. must be single-valued.
11 Postulate 1: The Physical Meaning of the Wavefunction
The Properties of the Wavefunction ( Y or )
(i) Normalization
V x x E x
dx
d x
m
2 2 2
TISE
If is a solution to this equation, than N is also a solution, where N is any
normalization constant.
For the normalized wavefunction N , the probability of finding the particle in the
region of space dx is given by (N )* (N )dx.
The sum over all space of these individual probabilities must equal 1:
- 1 2 N dx What is the value of the normalization constant, N? (*complex conjugate)
Postulate 2: Every Observable has a Corresponding Operator For every measurable property of the system, there is a corresponding quantum mechanical operator. An operator is a mathematical quantity that tells us to perform an operation: n n n O a ˆ
V ^ x ^ ^ x ^ E ^ x
dx
d x
m
n n n
2 2 2
TISE
Hamiltonian operator: ^
V x dx d m H 2 2 2 2 ˆ H x E x n n n ˆ This is an example of an eigenfunction/eigenvalue equation.
Postulate 2: Every Observable has a Corresponding Operator Common operators in quantum mechanics: Examples: (1) Momentum (2) Kinetic Energy (3) Potential Energy of the Harmonic Oscillator (4) Hamiltonian
16 Postulate 2: Every Observable has a Corresponding Operator Some quantum mechanical operators do not generate eigenfunction/eigenavlue equations. 2 A cos kx Example: What is the momentum of a particle moving in 1D? Assume the following form of the wavefunction: ( ) ikx ikx A e e To find the linear momentum we must operate on the wavefunction with : x p ˆ
A kx
dx
d
i
p ˆ^ x 2 cos
A kx
i
kx
dx
d
A
i
2 sin
2 cos
In general, when is not an eigenfunction of an operator, the property to which the
operator corresponds does not have a definite value. Recall ( ) 1 cos ikxikx x e e
Postulate 3: The Result of any Individual Measurement
In any single measurement of the observable that corresponds to the operator O ˆ ,
the only values that will ever be measured are the eigenvalues of that operator. H x t E x t n n n , , ˆ Y Y Solving the Schrödinger equation means finding the complete set of eigenfunctions
(Y n ) and eigenvalues ( En ) of the Hamiltonian operator.
Some properties of a complete set: k n n k k c c c c 1 1 2 2
- A set of functions is complete when any arbitrary function can be expressed as a superposition (linear combination) of them:
- In a single measurement, only one of the eigenvalues corresponding to
the particular k that contributes to the superposition will be found.
Postulate 4: The Expectation Value If the system is in a state described by the wavefunction Y, then the average value of a large number of observations is given by the expectation value of the operator of interest: Y Y Y Y d O d O
ˆ
ˆ For example, the average kinetic energy for a particle moving in 1D: E E d k k ˆ
dx dx d m 2 2 2
2
for normalized
Variance:
The time-evolution of a quantum mechanical system is governed by the time- dependant Schrödinger equation. Postulate 5: The Time Evolution of a Quantum Mechanical System t x t i H x t Y Y , , ˆ For one particle moving in 1D: i ^ Et x t x e Y ,