Inner Product Practice
Consider two column vectors (a.k.a kets in quantum mechanics) ∣ α ⟩ = ( 3 , − i , 2 − i ) and ∣ β ⟩ = ( 2 , i , 2 − i ) .
Calculate ⟨ α ∣ β ⟩ .
The answer is 8.
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1 solution
Yes, you are right.
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Funny how your "solution" achieved the correct answer though! :)
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Jake, it was just an coincident. I misread the question. Missing the comma. I learn bra-ket in my final year of Engineering in University of Malaya more than 30 years ago. I have never in touch with it ever since. How come a 15-year-old learn about it so early. Self-interest has driven you to learn about it?
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@Chew-Seong Cheong – Yup. I was interested in vector spaces and vectors in general, so I'm familiar with bra-ket and inner products.
How (-i * i) = -1 ? (-i * i) = (-i²) = (- -1) = +1
How (sqrt(2) -i) (sqrt(2) -i) = 3 ? (sqrt(2) -i) (sqrt(2) -i) = (sqrt(2)² - 2 sqrt(2) i + i²) = (2 - 2 sqrt(2) i - 1) = (1 - 2 i sqrt(2)) = (1 - 2*sqrt(-2))
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I think you are talking about − i ⋅ i , there is a line on top. − i is the conjugate of − i , which is i . Therefore,
− i ⋅ i = i ⋅ i = − 1 .
Similarly, ( 2 − 1 ) ⋅ ( 2 − i ) = ( 2 + 1 ) ( 2 − i = 3
⟨ α ∣ β ⟩ = 3 ⋅ 2 + − i ⋅ i + ( 2 − i ) ⋅ ( 2 − i ) = 6 + ( − 1 ) + ( 2 + 1 ) = 8