Rewrite state in new basis - Quantum Mechanics

[12x4]

Homework Statement

Rewrite the state |ψ⟩ = √(1/2)(|0> + |1>) in the new basis.

|3⟩ = √(1/3)|0⟩ + √(2/3)|1⟩

|4⟩ = √(2/3)|0⟩ − √(1/3)|1⟩

You may assume that |0⟩ and |1⟩ are orthonormal.

Homework Equations

The Attempt at a Solution

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I have a similar example in my notes however there is a step that I has stumped me. Annoyingly its the first one.

In my notes I have:

"""If we want to work in the basis |+⟩ and |−⟩ instead of | ↑⟩ and |↓⟩, with,

|+⟩ = (1/√2)(| ↑⟩ + | ↓⟩) & |−⟩ = (1/√ 2)(| ↑⟩ − | ↓⟩)

how would |ψ⟩ and I be written in the new basis?

Let us rearrange as:

| ↑⟩ = 1/(√2)(|+⟩ + |−⟩) & | ↓⟩ = (1/√2)(|+⟩ − |−⟩)"""

After rearranging I think that I should be able to complete the question but as it stands I can't see how to rearrange them to get |0> & |1>. Any advice would be much appreciated as really struggling with Dirac notation at the moment. Thanks 12x4

 

[vela]

You're probably just getting confused by the new notation. Consider the ordinary algebraic equations
\begin{align*}
u &= \frac{1}{\sqrt 2} x + \frac{1}{\sqrt 2} y \\
v &= \frac{1}{\sqrt 2} x - \frac{1}{\sqrt 2} y
\end{align*} How would you solve for $x$ in terms of $u$ and $v$? You can essentially do the same thing.

 

[12x4]

thanks vela, I just managed to do it with your advice.