Find Vector Representation of |ψi> in |ei> basis

[jasonchiang97]

Homework Statement

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

Homework Equations

|ψ_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 |ψ_i> = 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

 

[DrClaude]

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.

 

[jasonchiang97]

Wouldn't the bra just be

(c₁ c₂) as you can just take the transpose of the matrix?

 

[DrClaude]

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.

 

[jasonchiang97]

Ah I see. Thanks!

 

[nrqed]

Homework Statement

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

Homework Equations

|ψ_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 |ψ_i> = 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>