Transcript Hamiltonians and Quantum Mechanics

Hamiltonians and Quantum Mechanics
• The order of measuring momentum and
position, for example, is arbitrary in large
systems.
• In small systems, the measurement of
momentum affects position, so the order is
important.
• Physicists in the early 20th Century were looking
for non-commuting systems
• matrices
• derivatives
Hamiltonians and Quantum Mechanics
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Physical quantities are represented by operators
Physical states are represented by wavefuntcions
Operator (w.f.) = number × (w.f.)
Number is the value measured for the physical
quantity when the system is in the specific state
• Matter like light:
𝐸 = ħ𝜔
𝑝 = ħ𝑘
ψ = 𝐴𝑒 𝑖(𝑘𝑥−𝜔𝑡)
𝜔
𝑐=
𝑘
Hamiltonians and Quantum Mechanics
• Momentum operator
𝜕
𝑝 = −𝑖ħ
𝜕𝑥
𝜕
𝑝 = −𝑖ħ 𝐴𝑒 𝑖(𝑘𝑥−𝜔𝑡) = ħ𝑘𝐴𝑒 𝑖(𝑘𝑥−𝜔𝑡)
𝜕𝑥
• Energy operator
𝜕
𝐸 = +𝑖ħ
𝜕𝑡
𝜕
𝐸 = +𝑖ħ 𝐴𝑒 𝑖(𝑘𝑥−𝜔𝑡) = ħ𝜔𝐴𝑒 𝑖(𝑘𝑥−𝜔𝑡)
𝜕𝑡
Hamiltonians and Quantum Mechanics
• Kinetic energy operator
𝑝2
ħ2 𝜕 2
𝑇=
=−
2𝑚
2𝑚 𝜕𝑥 2
• Potential energy operator
𝑈 = 𝑈(𝑥)
• Hamiltonian operating on wavefunction
ħ2 𝜕 2 ψ
𝐻ψ = 𝑇ψ + 𝑈ψ = −
+ 𝑈(𝑥)ψ
2
2𝑚 𝜕𝑥
Schrödinger’s Equation
𝐻ψ = 𝐸ψ
ħ2 𝜕 2 ψ
𝜕ψ
−
+ 𝑈 𝑥 ψ = 𝑖ħ
2
2𝑚 𝜕𝑥
𝜕𝑡
or
ħ2 𝜕 2 ψ
−
+ 𝑈 𝑥 ψ = 𝐸ψ
2
2𝑚 𝜕𝑥
Eigenvalue problem: Find E and ψ.
Physics 321
Hour 21
Hamiltonian Examples
The Hamiltonian - Notes
• The Hamiltonian is a function of p and q. But p is
not ‘the momentum,’ it is the generalized
momentum conjugate to q.
𝜕ℒ
𝑝𝑖 =
𝜕𝑞𝑖
• The general expression is:
𝐻=
𝑝𝑖 𝑞𝑖 − ℒ
𝑖
• That means you generally have to find the
Lagrangian before you can find he Hamiltonian!
The Hamiltonian
• 𝐻 = 𝐻(𝑞, 𝑝, 𝑡) whereas ℒ = ℒ(𝑞, 𝑞, 𝑡)
• For simple systems, H = T + U
• The equations of motion are first order:
𝜕𝐻
= −𝑝
𝜕𝑞
𝜕𝐻
= +𝑞
𝜕𝑝
Example
Hamiltonian2.nb