Question 110 - When an algebraic product is defined on the space, the Lie bracket is the commutator [x,y] = xy - yx according to Lie algebra in mathematics. If [p,x] f(x) = px f(x) - xp f(x), p = -ih d / dx, find the value of [p,x] in term of i and h.
Answer 110 - Let [p,x] f(x) = px f(x) - xp f(x) = (-ih d / dx) x f(x) - x (-ih d / dx) f(x) = -ih { d / dx [ x f(x) ] - x d f(x) / dx } = -ih [ f(x) ] + x d f(x) / dx - x d f(x) / dx = -ih [ f(x) ]. Then [p,x] = -ih. The answer is given by Kang Chuen Tat; PO Box 6263, Dandenong, Victoria VIC 3175, Australia; SMS +61405421706; chuentat@hotmail.com; http://kangchuentat.wordpress.com.
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X is strong but has a very low density (1% of traditional earth materials.) and hence light weight. It is a recyclable material. The compression behaviour of X is strain rate dependent. Higher strain rates result in higher initial modulus and higher compression strength. It can also withstand unlimited number of cycling loading provided the repetitive loads are kept below 80% of the compressive strength. The internal structure of the material includes air-traps which make it poor heat conductor. X is non- biodegradable and chemically inert in both soil and water. Most acids and their water solutions do not attack it; however strong oxidizing acids do. Solvents which attack X include esters, ketones, ethers, aromatic and aliphatic hydrocarbons and their emulsions, among others. It does not support bacterial/fungal growth as well .It also has significant acoustic properties and effectively reduces the transmission of airborne sound. X is combustible and should not be exposed to open flame or other ignition sources. Combustion products are carbon monoxide, carbon dioxide, water and soot. Long-term exposure to sunlight causes yellowing and a slight embrittlement of the surface due to ultraviolet light. X is able to withstand the rigours of temperature cycling, assuring long-term performance.
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