Coefficients of a particle in Linear superposition (QM)

[Johnahh]

Homework Statement

a particle is in a linear superposition of two states with energy E_0 \ and\ E_1
|\phi> = A|E_0> + \frac{A}{(3-\epsilon)^{1/2}}|E_1>

where:
A \ > \ 0, \ 0\ <\ \epsilon \ <\ 3

What is the value of A expressed as a function of epsilon

Homework Equations

P(E_0) \ +\ P(E_1) = 1\\<br /> P(E_0) = |&lt;E_0|\phi&gt;|^2\\<br /> P(E_1) = |&lt;E_1|\phi&gt;|^2

The Attempt at a Solution

My attempt was to normalise the function to find a value for A in terms of epsilon.
&lt;E_0|\phi&gt; = A\\<br /> &lt;E_1|\phi&gt; = \frac{A}{\sqrt{(3-\epsilon)}} \\<br /> |&lt;E_1|\phi&gt;|^2 + |&lt;E_0|\phi&gt;|^2 = A^2 + \frac{A^2}{(3-\epsilon)} = 1\\<br /> A^2(1 + \frac{1}{3-\epsilon}) = 1\\<br /> 4A^2 -\epsilon A^2 = 3-\epsilon\\<br /> A^2 = \frac{3-\epsilon}{4-\epsilon}\\<br /> A = \sqrt\frac{3-\epsilon}{4-\epsilon}\\

But this does not give me a value of 1 when i put it back in? I'm unsure where I am going wrong - only had 2 QM lectures so my knowledge is limited.

 

[blue_leaf77]

But this does not give me a value of 1 when i put it back in?

I think your answer is correct.

 

[Johnahh]

I think your answer is correct.

Wow, I was being an absolute idiot, and inputting A back in incorrectly, for some reason i was just putting the value of A back into
&lt;E_0|\phi&gt; and &lt;E_1|\phi&gt; expecting to get one.

Thankyou lol