Oct 12, 2016 · Obviously in both cases we get the zero vector because remember $\norm x = 0 \iff x =0$. However there is an ambiguity about the direction of the zero vector. Since both of the vectors were showing in different directions and a vector (in this a the zero vector) should show in only one direction.

Jun 29, 2024 · $\begingroup$ Without the zero vector, you lose the vector space structure of the Hilbert space. So even if it does not appear in the final step of calculating an expectation value or a probability amplitude, you absolutely need it to justify the calculations leading up to them, or to even make the theory mathematically consistent. $\endgroup$

Apr 20, 2020 · Your doubt is legitimate. In fact the proper way to define elements of such space is via equivalence classes. An element of such space is strictly speaking not a function but an equivalence class of functions that differ on a set of measure zero. The zero vector is the equivalence class of functions that are zero almost everywhere.

Jun 10, 2023 · It has to be projected onto $\text{span}\{|0\rangle\}$, but it's orthogonal to that subspace and such a projection produces the zero vector. That doesn't make any sense, because quantum states are rays (or equivalently unit vectors) in the Hilbert space.

The first is the zero vector of the vector space—the one you study in your linear algebra courses, the one which is the identity element of the vector addition operation. The second, on the other hand, is completely different: the zero inside the ket is just a label of the corresponding eigenvalue (of the number operator).

so that this state has the lowest possible non-zero energy, $$ \hat{H}|0\rangle = \frac{1}{2}\hbar\omega|0\rangle $$ Edit: The zero vector is just a mathematical object with no physical interpretation. The zero vector is formally defined as the additive identity of the additive group so that, $$\mathbf{u} + \mathbf{0} = \mathbf{u}$$

This distinction might be more clear by considering, for example, the inner product of a row and column vector in component form. For a zero vector, the components of the vector are zero. But, the components exist and so, the inner product can be formed. For a point, there are no components, no way to form the inner product with, e.g., a basis ...

Jun 11, 2015 · $\begingroup$ Vector area will ofcourse be zero.for a small area of the closed surface there will be vector in the outward direction.Now,for the whole surface there will be infinite no of such vectors which cancel out and gives us a zero vector. but In Gauss law you are taking surface integral of electric field which is a quite different case ...

Jun 13, 2016 · A vector is a quantity described by a magnitude and a direction. The magnitude is always +ve or zero. A -ve sign in front of a vector indicates the same magnitude but in the opposite direction. The - sign is part of the direction rather than the magnitude. Like all vectors, a "resultant vector" is neither +ve or -ve.

May 16, 2019 · The magnitude of the 4-velocity of light is always zero (see my comment above). The (always) non-zero time component of the 4-velocity of light does NOT mean that light moves in time. To calculate the 4-velocity of light, we have to use a different affine parameter instead of proper time, because the proper time of light is always zero.