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20 lines
564 B
Plaintext
20 lines
564 B
Plaintext
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Translation Operators
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Matrix mechanical expression:
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T̂(a𝓍 î) ❙x,y,z❭ = ❙x+a𝓍,y,z❭
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T̂(a𝓎 î) ❙x,y,z❭ = ❙x,y+a𝓎,z❭
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T̂(a𝓏 î) ❙x,y,z❭ = ❙x,y,z+a𝓏❭
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Position space form:
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T(a𝓍 î) = exp(-ι p𝓍 a𝓍/ħ )
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Translation operators commute since momentum operators commute.
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But, [x̂ᵢ,p̂ⱼ]=ιħδᵢⱼ
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T̂(𝐚) = exp(-ι p𝓍 a𝓍/ħ ) exp(-ι p𝓎 a𝓎/ħ ) exp(-ι p𝓏 a𝓏/ħ )
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= exp(-ι 𝐩 𝐚/ħ )
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❬r❙p̂❙Ψ❭ = ħ/ι ∇ ❬r❙Ψ❭
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