The states `|psi_1>` and `|psi_2>` will be orthogonal if and only if the product `<psi_1|psi_2> = 0` .Therefore, to find the value of x that makes the states orthogonal, we will need to substitute the definitions of the functions into the equation
`<psi_1|psi_2> = 0`
and solve the resulting equation for x.
`|psi_1> = |s> + 2i |t>` and
`|psi_2> = 2|s> + x|t>`
The complex conjugate `<psi_1|` ` `
is `<psi_1| = <s| -2i<t|`
So `<psi_1|psi_2> = (1)(2) <s|s> + (-2i)(x) <t|t> = 0`
Because `|s>` and `|t>` are orthonormal states, we know that by definition
`<s|s> = <t|t> = 1` so
`2 - 2ix = 0`
`2ix = 2` and
`x = 1/i` .
We have to be careful with - signs.
Since `i = sqrt(-1)`
We know `i^2 = -1`
and so `-i^2 = 1` .
Thus `x = -i^2/i`
or `x =-i` , and
`|psi_2> = 2 |s> - i |t>` .
Substituting this value into the equation above, we find
`<psi_1|psi_2> = 2 - 2 = 0` so the value of `-i` does cause the states to be orthogonal.