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.

We have

`|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.