Lecture 3 - Energyandtime Hamiltonian

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Energy and time
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Hamiltonian 2 2 ˆ () 2 HV m =− ∇ + r = Classical energy in quantum mechanics cam also be converted to an operator called Hamiltonian 22 ˆ pp E Ux H Vx mm =+ In order to define the explicit form of this operator we need to define the square of an operator. 2 2 222 2 2 ˆˆ ˆ ˆ ˆ ; ˆˆˆ xy xyz A f xA A f A f x pf i i f f pf i i f f xx x yy y ppp ⎡⎤ == ⎣⎦ ⎛⎞ ∂∂ ⎜⎟ ⎝⎠ =++= p = = = Compare this definition with Schrödinger equations ( ) 2 2 , , 2 , ˆ t iU t tm t iH t ∂Ψ ∇ Ψ+ Ψ ∂Ψ r rr r = = =
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Time independent potential ( ) () 2 , , 2 t iV t tm ∂Ψ ⎛⎞ =− Δ + Ψ ⎜⎟ ⎝⎠ r rr = = If the potential does not depend on time we can solve this equation by separating variables: ( ) ,( ) ( ) tt ψ ϕ Ψ≡ Substitute it to the equation: 2 2 2 ; 2 2 )( 2 ( ) dt it V dt m tV i m dt V i m dt t ψϕ ϕψ λ Δ + −Δ + = + == r r r = = = = = =
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Time dependent part () i t dt i dt it td t i dt t Ce t λ ϕ λλϕ ϕλ = ⇒= =− = = = = =
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Time independent part 2 2 () 2 2 ˆ V m V m H ψ λψ λ ψλ ⎛⎞ −Δ + ⎜⎟ ⎝⎠ =⇒ Δ + = = rr r r = = Stationary (time independent) Schrödinger equation has a form of eigenfunction equation, where the separation constant plays the role of the eigenvalue
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The meaning of the separation constant Calculate the average energy and its uncertainty in one these states assuming that the wave functions are normalized () *3 * 3 2* 3 * 3 2 * 3 2 22 2 ˆˆ ,, ( ) ( ) ( ) ( ) (wave function is normalized) ( ) ( ) ˆ ()() 0 ii tt E Et H t d r e H e d r dr H H t d r e e H H d r Hd r d r EE λλ ϕϕ λϕ ϕ λ λ ϕ σ ⎡⎤ Ψ = = ⎢⎥ ⎣⎦ = Ψ = = == =−= ∫∫ rr r r r r r r Thus, in a stationary state a system has a definite value of energy, and λ should be identified with it. This is the second situation illustrating a general rule: if a system is in a state, which is eigen state of an operator, then the quantity whose this operator represents has a definite, not random value. Identifying λ with energy we assume that it is real
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Stationary States () ,( ) i Et E te ψ Ψ= rr = Average values of all physical quantities also do not change with time in a stationary state, which is very easy to see: *3 ˆˆ (, ) () (, ) () (, ) () ii Et Et Qe Q e d r Qd
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This note was uploaded on 09/27/2010 for the course PHYSICS 365 taught by Professor Deych during the Spring '10 term at SUNY Stony Brook.

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Lecture 3 - Energyandtime Hamiltonian

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