Simple vectors in QM

  • #1

Homework Statement



Consider the following ket: |ψi> = c1|e1> + c2|e2>, where ci are some complex coefficients. Find the column-vector representation of |ψi> in the |ei> basis. Find the row-vector representation of <ψ| in the <ei| basis.

Homework Equations


i> = c1|e1> + c2|e2>

The Attempt at a Solution



Well I'm not sure what to do so I tried to start off by solving c1 and c2. To do this I multiplied |ψi> = c1|e1> + c2|e2> by <e1| to get that c1 = <e1|ψ> . Multiplying the same equation by <e2| gives c2 = <e2|ψ>

So I wrote |ψ> = <e1|ψ|e1> + <e2|ψ|e2>

Now I'm not sure what to do
 
Last edited:

Answers and Replies

  • #2
DrClaude
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The question asks for you to write it as column and row vectors, i.e.,
$$
\begin{pmatrix} a \\ b \end{pmatrix}
$$
and ##(\alpha, \beta)##.

I'll give you the answer for the first part, since you are almost there and there is no fundamental principle here, just a convention. Basically, you simply need to arrange your ##c_i## in a vector, so for ##|\psi_i\rangle##, you get the representation
$$
|\psi_i\rangle \doteq \begin{pmatrix} \langle e_1 | \psi_i \rangle \\ \langle e_2 | \psi_i \rangle \end{pmatrix} = \begin{pmatrix} c_1 \\ c_2 \end{pmatrix}
$$
(with ##\doteq## meaning "is represented by," as the representation is not unique and will depend on the basis used and the order of the base states in that basis).

I'll let you figure out what the answer looks like for a bra.
 
  • #3
Wouldn't the bra just be

(c1 c2) as you can just take the transpose of the matrix?
 
  • #4
DrClaude
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Wouldn't the bra just be

(c1 c2) as you can just take the transpose of the matrix?
Not exactly. The bra is not equivalent to the transpose of the ket, but to its Hermitian transpose.
 
  • #6
nrqed
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Homework Statement



Consider the following ket: |ψi> = c1|e1> + c2|e2>, where ci are some complex coefficients. Find the column-vector representation of |ψi> in the |ei> basis. Find the row-vector representation of <ψ| in the <ei| basis.

Homework Equations


i> = c1|e1> + c2|e2>

The Attempt at a Solution



Well I'm not sure what to do so I tried to start off by solving c1 and c2. To do this I multiplied |ψi> = c1|e1> + c2|e2> by <e1| to get that c1 = <e1|ψ> . Multiplying the same equation by <e2| gives c2 = <e2|ψ>

So I wrote |ψ> = <e1|ψ|e1> + <e2|ψ|e2>

Now I'm not sure what to do
Watch out, the last expression you wrote is nonsensical because you have a ket on the left and a number on the right. You really meant
|ψ> = <e1|ψ> |e1> + <e2|ψ> |e2>
 
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