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By Edward B. Stuart, Alan J. Brainard, Benjamin Gal-Or

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Dx PROBLEM 8 Classical electromagnetic theory uses real magnetic and electric fields coupled via Maxwell’s equations. The magnetic and electric fields each have physical meaning. Both fields are needed to describe both the instantaneous state and time evolution of the Applied quantum mechanics 3 system. Quantum mechanics uses one complex wave function to describe both the instantaneous state and time evolution of the system. It is also possible to describe quantum mechanics using two coupled real wave functions corresponding to the real and imaginary parts of the complex wave function.

For the operator xˆ this is easy to show ∞ 〈ψ| xψ 〉 = ∞ ∫ψ * ( x ) ( xψ ( x ) ) d x = –∞ ∫ ( xψ ∞ * ( x )) ψ( x )d x = –∞ ∫ ( xψ ( x ) ) * ψ ( x ) dx = 〈x ψ|ψ 〉 –∞ d is anti-Hermitian we integrate by parts and make use dx of the fact that the wavefunction ψ ( x ) → 0 as x → ±∞ (b) To show that the operator d ψ〉 = dx 〈 ψ| ∞ * *  d  ∫ ψ ( x )  ih d x ψ ( x )  d x = ψ ( x )ψ ( x ) –∞ x=∞ x = –∞ ∞ – d ∫ dx ψ * –∞ (x )ψ (x )d x  ∞ 〈 ψ| * d d d ψ 〉 = – ∫  ψ ( x ) ψ ( x ) dx = – 〈 ψ| ψ〉 dx d x d x –∞ † d d and   = – .

0 ∞ ∞ 1d 1d 2 2 〈 p r〉 = – ih ∫ φ *100 -- ( r φ 100 ) 4πr d r = – ih ∫ φ 100 -- ( rφ 100 ) 4πr dr = – ihA r r d r d r 0 0 where A is a real number. , H = H† and 〈ψ |Hψ 〉 = 〈 Hψ|ψ 〉 . , xˆ † = xˆ * or 〈ψ |x | φ〉 = 〈φ |x |ψ〉 . For the operator xˆ this is easy to show ∞ 〈ψ| xψ 〉 = ∞ ∫ψ * ( x ) ( xψ ( x ) ) d x = –∞ ∫ ( xψ ∞ * ( x )) ψ( x )d x = –∞ ∫ ( xψ ( x ) ) * ψ ( x ) dx = 〈x ψ|ψ 〉 –∞ d is anti-Hermitian we integrate by parts and make use dx of the fact that the wavefunction ψ ( x ) → 0 as x → ±∞ (b) To show that the operator d ψ〉 = dx 〈 ψ| ∞ * *  d  ∫ ψ ( x )  ih d x ψ ( x )  d x = ψ ( x )ψ ( x ) –∞ x=∞ x = –∞ ∞ – d ∫ dx ψ * –∞ (x )ψ (x )d x  ∞ 〈 ψ| * d d d ψ 〉 = – ∫  ψ ( x ) ψ ( x ) dx = – 〈 ψ| ψ〉 dx d x d x –∞ † d d and   = – .

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