| Concept | Abstract notation | Ordinary vectors | Matrices | Wave Functions [square well 0 to a] |
|---|---|---|---|---|
| Vector | |a>, |b>, ... | r | (a) (b) (c) |
y(x) |
| Scalar |
a, b, ... In general, the complex numbers |
|||
| Addition | |a> + |b> = |g> | r1 + r2 = r3 | (a) (d) (a+d) (b) + (e) = (b+e) (c) (f) (c+f) |
y1(x) +y2(x) = y3(x) |
| Zero vector | |a> + |0> = |a> |a> + | -a> = |0> |
0 | (0) (0) (0) |
0 |
| Scalar multiplication | a |a> = |g> | |||
| Linear combination of vectors | a |a> + b |b> + c |g> + ... | |||
| Linear independence | a |a> + b |b> + c |g> + ... = |0> implies a = b = c = ...=0 | |||
| Basis | A set |e1> , |e2> , ... of linearly-independent vectors that span the space | 1x, 1y, 1z | (1) (0) (0) (0) , (1) , (0) (0) (0) (1) |
sin (npx/a) |
| Components | if |a> = Sn an |an> , the set of coefficients (a1 , a2 , ...) | rx , ry , rz | Obvious | If y(x) = Sn ansin(npx/a), the set (a1 , a2 , ...,) |
| Concept | Abstract notation | Ordinary vectors | Matrices | Wave Functions [square well 0 to a] |
|---|---|---|---|---|
| Inner product | <a | b> =
<b | a>* <a | a> ³ 0 and <a | a> = 0 <=> | a> = | 0>
<a|
[b | b >
+ c | g>] | r1 . r2 | Sn an* bn | ò dx y1*(x) y2(x) |
| Norm | ||a|| = Ö <a | a> | Ör2 | Ö Sn |an|2 | Ö ò dx |y(x)|2 |
| Orthonormal set | <en | em> = dnm | See bases above | ||
| Complete set | Sn | en >< en | = 1 | |||
| Last Revised 01/09/26 |