QUANTUM PHYSICS QUESTION (Best answer gets many points)?
ok lets say i have a state vector that i want to convert from a position basis to another basis (maybe momentum?)
how do i do that
i know there is this formula in the quantum physics for dummies book but i dont get it
state vector: column matrix where you store your values
- AdamLv 44 years agoFavorite Answer
I also have the quantum physics for dummies book. I don't particularly like the book. Many colleges use a book called Introduction to Quantum Mechanics by David J. Griffiths. Anyways, were you looking at the formula on page 30? To understand that formula, you need to understand vectors and inner products.
| ψ > = ∑ | φi > < φi | ψ >
| φi > are unit vectors
< φi | ψ > is the inner product of | ψ > and | φi >. I'm going to call this the dot product because you have probably heard of the dot product before. From school, if you dot product a vector A and a vector B, you get a scalar. The scalar has the value |A||B|cosθ. If you dot product a vector A with a unit vector U, you get a scalar, |A|cosθ because the unit vector U is of length 1. If you want to write out a vector, you have to specify "how long it is" along some directions. So if vector A is 4 units long in the x direction and 5 units long in the y direction, you can write that, A = 4 i-hat + 5 j-hat, where i-hat and j-hat are unit vectors. How do I know that A is 4 units long along the i-hat direction? Because the dot product of (A ▪ i-hat) = 4. That is, (A) projected onto i-hat is 4 units long. Likewise, (A ▪ j-hat) = 5. (A) projected onto j-hat is 5 units long. But we don't have to represent vector A with respect to i-hat and j-hat directions. We can choose whatever basis we want. Maybe that same vector A is 1.5 units long along a u-hat direction and 3 units long along a v-hat direction. That is A projected onto u-hat, or (A ▪ u-hat), is 1.5 and A projected onto v-hat, or (A ▪ v-hat), is 3.
A = 1.5 u-hat + 3 v-hat = (A ▪ u-hat) u-hat + (A ▪ v-hat) v-hat.
This is what the equation you are looking at says. < φi | ψ > is like the dot product from school between the unit vector | φi > and | ψ > . This tells you how long | ψ > is along | φi >. So to change basis, you need to project your state vector onto the basis to see how long it is in those basis vector directions. Then you can neatly write out that (vector) = (this long) (in this unit direction) + (this long)(in this other unit direction) + etc. Or you can write the vector as a column matrix (just as long as you know which directions correspond to which element of the matrix it's all the same thing. Just different representations of the same vector)