Inner product, dot product?

• I
• lawlieto
In summary, the inner product is defined differently in different sources, but the main properties remain the same, such as linearity and the use of conjugates for complex components.

lawlieto

I started learning quantum, and I got a bit confused about inner and dot products.
I've attached 2 screenshots; 1 from Wikipedia and 1 from an MIT pdf I found online.

Wikipedia says that a.Dot(b) when they're complex would be the sum of aibi where b is the complex conjugate.

The PDF from MIT (https://ocw.mit.edu/courses/physics...all-2013/lecture-notes/MIT8_05F13_Chap_04.pdf) page 2 says that the inner product is taken <a|b>, then when doing a1b1+a2b3+... the complex conjugate of a is taken. I thought this would be the same thing as taking the dot product with complex numbers (like what I mentioned above in the 2nd paragraph). But in the dot product, the complex conjugate of b is taken, whereas here the complex conjugate of a is taken.

Could someone demystify this for me please?

Thanks

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lawlieto said:
I started learning quantum, and I got a bit confused about inner and dot products.
I've attached 2 screenshots; 1 from Wikipedia and 1 from an MIT pdf I found online.

Wikipedia says that a.Dot(b) when they're complex would be the sum of aibi where b is the complex conjugate.

The PDF from MIT (https://ocw.mit.edu/courses/physics...all-2013/lecture-notes/MIT8_05F13_Chap_04.pdf) page 2 says that the inner product is taken <a|b>, then when doing a1b1+a2b3+... the complex conjugate of a is taken. I thought this would be the same thing as taking the dot product with complex numbers (like what I mentioned above in the 2nd paragraph). But in the dot product, the complex conjugate of b is taken, whereas here the complex conjugate of a is taken.

Could someone demystify this for me please?
In the MIT materials they are defining an inner product as they show in equation 1.4. There are many ways that an inner product can be defined, including the one you show from the Wikipedia article.

lawlieto
Mark44 said:
In the MIT materials they are defining an inner product as they show in equation 1.4. There are many ways that an inner product can be defined, including the one you show from the Wikipedia article.

Thanks for your reply, so it's just a matter of how you define it? But then you could define anything to be anything?

lawlieto said:
Thanks for your reply, so it's just a matter of how you define it? But then you could define anything to be anything?
No, it has to satisfy some properties like linearity in its arguments. Another requirement is usually, that ##\langle a,a \rangle \geq 0##. In the case of complex components, this is the reason for the conjugate in either of the arguments. Whether you choose the first or second doesn't matter, just don't confuse them.

Although not required, it is customary to take the complex conjugate of the right hand term.