Expressing Vector w/o Basis: Dirac Bra-Ket Notation

In summary, the conversation discusses the concept of expressing a vector without reference to a basis. It is argued that even with the use of Dirac bra-ket notation, a vector must ultimately be expressed in a specific basis. This is similar to how in Euclidean vectors, specifying the length and angle still requires a basis to define a particular direction. The concept is further applied to the commutator between operators and the derivation of eigenvalues and eigenvectors. Ultimately, it is concluded that the wave function is a way to express other vectors in terms of the first vectors in the universe.
  • #1
daudaudaudau
302
0
Inspired by the Dirac bra-ket notation I came to think that an ordinary Euclidean vector must be expressible without reference to a basis. But if I specify the length and angle of a vector, I have to refer this angle to some particular direction. Isn't this the same as choosing a basis?

Edit: Well I guess length+angle is just polar coordinates ...
 
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  • #2
Even with bra-ket notation you don't really express a vector without a basis. You just attach some letter to it [tex] | \Psi > [/tex]. Think about it, does it really has any meaning to you? At some point you will have to express it some how (via some basis) to really work with it.

This is the same way with euclidean vectors. You can write something like [tex] \vec{v} [/tex] forever, and talk about it "interactions" with other anonymous [tex]\vec{u}[/tex]'s but what does that give you until you write it down in some basis? (And defining it through interaction with other vectors is just like selecting a basis).
 
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  • #3
If we have the commutator between operators [itex]\hat x[/itex] and [itex]\hat p[/itex] (position and momentum), we can derive the eigenvalues of the operator [itex]\hat x[/itex] and label the corresponding eigenvectors using the eigenvalues. But what are these eigenvectors REALLY? What do they look like? We know nothing about them, because they are sort of the first vectors in the universe. How will you express them in a basis, if you have no other vectors? But we can express other vectors as linear combinations of them, and this leads to the wave function. Does this make sense?
 

Related to Expressing Vector w/o Basis: Dirac Bra-Ket Notation

1. What is Dirac Bra-Ket notation?

Dirac Bra-Ket notation, also known as Dirac notation or bra-ket notation, is a mathematical notation used in quantum mechanics to represent quantum states and operators. It uses the symbols | and < to represent the "ket" and "bra" vectors, respectively.

2. Why is Dirac Bra-Ket notation useful?

Dirac Bra-Ket notation is useful because it simplifies the mathematical representation of quantum states and operators. It allows for easy manipulation and calculation of these quantities without having to use complex matrix operations.

3. How do you express a vector without a basis using Dirac Bra-Ket notation?

To express a vector without a basis using Dirac Bra-Ket notation, you can use the symbol | followed by the vector name. For example, if the vector is named v, you would write it as |v>.

4. What is the difference between a ket vector and a bra vector?

A ket vector, denoted by |v>, represents a column vector in a vector space, while a bra vector, denoted by <v|, represents a row vector. In other words, a ket vector is a state vector, while a bra vector is the dual vector of the ket vector.

5. Can Dirac Bra-Ket notation be used for classical mechanics?

No, Dirac Bra-Ket notation is specifically designed for quantum mechanics and is not applicable to classical mechanics. Classical mechanics uses different mathematical operations and notation to represent states and operators.

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