Quantum Review

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Review of Quantum Mechanics 2.1 States and Operators A quantum mechanical system is defined by a Hilbert space, H, whose vectors, ¸ ¸ ¸ψ _ are associated with the states of the system. A state of the system is represented by the set of vectors e iα ¸ ¸ ¸ψ _ . There are linear operators, O i which act on this Hilbert space. These operators correspond to physical observables. Finally, there is an inner product, which assigns a co
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  Review of Quantum Mechanics 2.1 States and Operators A quantum mechanical system is defined by a Hilbert space, H , whose vectors,  ψ  are associated with the states of the system. A state of the system is represented bythe set of vectors e iα  ψ  . There are linear operators, O i which act on this Hilbertspace. These operators correspond to physical observables. Finally, there is an innerproduct, which assigns a complex number,  χ  ψ  , to any pair of states,  ψ  ,  χ  . Astate vector,  ψ  gives a complete description of a system through the expectationvalues,  ψ  O i  ψ  (assuming that  ψ  is normalized so that  ψ  ψ  = 1), which wouldbe the average values of the corresponding physical observables if we could measurethem on an infinite collection of identical systems each in the state  ψ  .The adjoint, O † , of an operator is defined according to  χ  O  ψ  =  χ  O †  ψ  (2.1)In other words, the inner product between  χ  and O  ψ  is the same as that between O †  χ  and  ψ  . An Hermitian operator satisfies O = O † (2.2)7  Chapter 2: Review of Quantum Mechanics  8while a unitary operator satisfies OO † = O † O = 1 (2.3)If  O is Hermitian, then e i O (2.4)is unitary. Given an Hermitian operator, O , its eigenstates are orthogonal,  λ   O  λ  = λ  λ   λ  = λ   λ   λ  (2.5)For λ  = λ  ,  λ   λ  = 0 (2.6)If there are n states with the same eigenvalue, then, within the subspace spanned bythese states, we can pick a set of  n mutually orthogonal states. Hence, we can usethe eigenstates  λ  as a basis for Hilbert space. Any state  ψ  can be expanded inthe basis given by the eigenstates of  O :  ψ  =  λ c λ  λ  (2.7)with c λ =  λ  ψ  (2.8)A particularly important operator is the Hamiltonian, or the total energy, whichwe will denote by H  . Schr¨odinger’s equation tells us that H  determines how a stateof the system will evolve in time. i ¯ h∂ ∂t  ψ  = H   ψ  (2.9)If the Hamiltonian is independent of time, then we can define energy eigenstates, H   E   = E   E   (2.10)  Chapter 2: Review of Quantum Mechanics  9which evolve in time according to:  E  ( t )  = e − i Et ¯ h  E  (0)  (2.11)An arbitrary state can be expanded in the basis of energy eigenstates:  ψ  =  i c i  E  i  (2.12)It will evolve according to:  ψ ( t )  =   j c  j e − i Ejt ¯ h  E   j  (2.13)For example, consider a particle in 1 D . The Hilbert space consists of all continuouscomplex-valued functions, ψ ( x ). The position operator, ˆ x , and momentum operator,ˆ  p are defined by:ˆ x · ψ ( x ) ≡ xψ ( x )ˆ  p · ψ ( x ) ≡ − i ¯ h∂ ∂xψ ( x ) (2.14)The position eigenfunctions, xδ  ( x − a ) = aδ  ( x − a ) (2.15)are Dirac delta functions, which are not continuous functions, but can be defined asthe limit of continuous functions: δ  ( x ) = lim a → 0 1 a √  πe − x 2 a 2 (2.16)The momentum eigenfunctions are plane waves: − i ¯ h∂ ∂xe ikx = ¯ hke ikx (2.17)Expanding a state in the basis of momentum eigenstates is the same as taking itsFourier transform: ψ ( x ) =   ∞−∞ dk ˜ ψ ( k )1 √  2 πe ikx (2.18)  Chapter 2: Review of Quantum Mechanics  10where the Fourier coefficients are given by:˜ ψ ( k ) =1 √  2 π   ∞−∞ dxψ ( x ) e − ikx (2.19)If the particle is free, H  = − ¯ h 2 2 m∂  2 ∂x 2 (2.20)then momentum eigenstates are also energy eigenstates:ˆ He ikx =¯ h 2 k 2 2 me ikx (2.21)If a particle is in a Gaussian wavepacket at the srcin at time t = 0, ψ ( x, 0) =1 a √  πe − x 2 a 2 (2.22)Then, at time t , it will be in the state: ψ ( x,t ) =1 √  2 π   ∞−∞ dka √  πe − i ¯ hk 2 t 2 m e − 12 k 2 a 2 e ikx (2.23) 2.2 Density and Current Multiplying the free-particle Schr¨odinger equation by ψ ∗ , ψ ∗ i ¯ h∂ ∂tψ = − ¯ h 2 2 mψ ∗ ∂  2 ∇ 2 ψ (2.24)and subtracting the complex conjugate of this equation, we find ∂ ∂t ( ψ ∗ ψ ) = i ¯ h 2 m  ∇·  ψ ∗   ∇ ψ −    ∇ ψ ∗  ψ  (2.25)This is in the form of a continuity equation, ∂ρ∂t =   ∇·   j (2.26)The density and current are given by: ρ = ψ ∗ ψ
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