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Introduction To QM

1) Classical mechanics describes how objects behave through motion, position, velocity, mass, energy, and acceleration. However, it fails to explain certain phenomena like black body radiation. 2) Quantum mechanics provides a more accurate description of nature at microscopic scales and addresses these unexplained phenomena through concepts like wave-particle duality and quantization of energy. 3) Quantum mechanics uses wave functions and differential equations to determine the properties and behavior of particles like position, momentum, and energy over time.

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66 views3 pages

Introduction To QM

1) Classical mechanics describes how objects behave through motion, position, velocity, mass, energy, and acceleration. However, it fails to explain certain phenomena like black body radiation. 2) Quantum mechanics provides a more accurate description of nature at microscopic scales and addresses these unexplained phenomena through concepts like wave-particle duality and quantization of energy. 3) Quantum mechanics uses wave functions and differential equations to determine the properties and behavior of particles like position, momentum, and energy over time.

Uploaded by

Nicholas Ow
Copyright
© Attribution Non-Commercial (BY-NC)
We take content rights seriously. If you suspect this is your content, claim it here.
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Download as DOCX, PDF, TXT or read online on Scribd
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CHEM 1301 | Introduction to Quantum Mechanics

Principles of CM

 Classical mechanics describes how objects such as particles and waves behaves

 It describes motion, position, velocity, mass, energy, acceleration etc. of the particle
Basic

Quantity Symbol Definition SI Unit


Position x m
Time t s
Mass m kg
Δ x x 1−x 2 d x
Velocity v v= = = = ẋ ms-1
Δ t t 1−t 2 d t
dv
Acceleration a a= = ẍ ms-2
dt
Momentum p p=mv=m ẋ m kg s-1
p2 1
Derived

2
Kinetic energy T T= = mv m2 kg s-2 = J
2m 2
Potential energy V J
Total energy E E=T +V J
−d v
Force F F= m kg s-2 = N
dx

 Note: velocity, acceleration, momentum and force are vectors (both magnitude and
direction)

 Force = rate of change of potential energy with position

 A free particle refers to a particle which has no force acting upon it and is neither
accelerating nor decelerating

 Thus, T > 0 (V = 0 or constant)

Harmonic Oscillator (spring)

1 2 p2
V x = Dx T=
( )
2 2m
0 x
 Where D = spring constant (the larger the D, the stiffer the spring)

 Object is moving the fastest at x = 0 (V=0 and E=T)

 At extreme ends of the curve, T=0

−d v −d 1
F=
dx
=
dx 2 ( )
D x 2 =−Dx (Hooke’s Law)
2

Newton’s Laws of Motion

1. A particle moves with constant velocity unless acted upon by a force

(If F=0, a=0, V is constant)

2. If acted upon by a force, a particle will change velocity (or momentum) (i.e.
accelerate)

(F = ma or F = dp/dt since p = mv and a = dv/dt)

Equations of Motion (EOM)

 Describes the trajectory of particles (i.e. the position as time changes)

 Solving the EOM gives the position of the particle after a time t

dp
Momentum ṗ= =F which when integrated gives p(t)
dt

dx p
Position ẋ= = which when integrated gives x(t)
dt m

 Can be solved given suitable initial / boundary conditions

Free particle

 Velocity is constant, so p = constant and dp/dt = 0 and dx/dt = p/m = constant

 Let the initial conditions be p(t=0)=p0 and x(t=0)=x0

p0 t p
From EOM, p ( t ) =p 0 x ( t ) =x0 + (by integrating∫ dx=∫ dt )
m m

Harmonic Oscillator

 Solutions are p ( t ) =−p 0 sin ( ωt ) x ( t )=x 0 cos ⁡(ωt)

D
 Initial conditions: p0 = 0 and x0 where ω=
√ m

Wave Mechanics

 Waves have wavelength  which has period T

 A wave can be described by the 2nd order partial differential equation wave equation

∂2 y 2
2 ∂ y
−c =0
∂ t2 ∂ x2

 A general solution can be obtained by setting initial conditions (x0,y0) giving


3

ω
y ( t ) = y 0 cos ( kx−ωt ) if c=
k

Important Equations

2 π where  = wavelength
Wave vector k =
❑

2π
Angular frequency ω= where T = vibrational period
T

ω
Wave velocity c=
k

dx ωt
Since c= then x=
dt k

Direction of waves

y ( t ) = y 0 cos ( kx−ωt ) (travelling right)

y ( t ) = y 0 cos (−kx−ωt ) (travelling left)

Standing Waves

y ( t ) = y 0 [cos ( kx−ωt ) +cos (−kx−ωt ) ]=2 y 0 cos ( ωt ) cos ⁡(kx )

 When cos(kx) = 0, there will be nodes (i.e. y = 0) for all values of t

❑
 Nodes will be at (n+1)
2 where n = 0, +1, +2, ...

Standing Waves in Boxes

 Boundary conditions are y(0) = 0 and y(L) = 0 (zero at both ends of the box)

Interference

 Waves an constructively (in-phase) and destructively (out-of-phase) interfere and


may also be diffracted

 The energy of the wave is proportional to its amplitude

Failure of CM

Black Body Radiation

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