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Wavefunction normalisation

  • Thread starter ParoxysmX
  • Start date
21
0
1. Homework Statement
Determine the constant λ in the wave equation

[itex]\Psi(x) = C(2a^2 x^2 + \lambda)e^{-(a^2 x^2/2)}[/itex]

where [itex]a=\sqrt{mω/\hbar}[/itex]

2. Homework Equations

Some standard integrals I guess

3. The Attempt at a Solution

So I believe the wave equation just needs to be normalised. Using the usual conditions for normalisation,

[itex](C2a^2 + C\lambda)^2 \int^{∞}_{-∞} | x^2 e^{-(a^2 x^2/2)} + e^{-(a^2 x^2/2)} |^2 dx =1[/itex]

From there,

[itex](C2a^2 + C\lambda)^2 \int^{∞}_{-∞} |2x^2 e^{-(a^2 x^2/2)}|^2 dx =1[/itex]

Then squaring the function inside the integral and moving the '4' outside the integral as it is a constant,

[itex]4(C2a^2 + C\lambda)^2 \int^{∞}_{-∞} x^4 e^{-(a^2 x^2)} dx =1[/itex]

Now that should be a standard integral but I don't know any involving an x term to the fourth power. Or perhaps I've done something else wrong?
 

Answers and Replies

DrClaude
Mentor
7,010
3,184
[itex](C2a^2 + C\lambda)^2 \int^{∞}_{-∞} | x^2 e^{-(a^2 x^2/2)} + e^{-(a^2 x^2/2)} |^2 dx =1[/itex]
Since when is
$$
[(ab+c) d]^2 = (a+c)^2(bd+d)^2
$$

The next step is also completely wrong. Start again from ##|\psi|^2 = \psi^* \psi##. You should also allow ##C## and ##\lambda## to be complex.
 
21
0
If the function doesn't contain any complex exponentials, then [itex]\psi^{*}[/itex] is the same as [itex]\psi[/itex], isn't it?
 
DrClaude
Mentor
7,010
3,184
As all physical observables depend ultimately on ##| \psi |^2##, the wave function of a physical system is only defined up to a complex phase. In other words, ##\psi## and ##\psi e^{i \delta}##, with ##\delta## real, decribe the same thing. Therefore, you can choose the normalization constant ##C## in
$$
\psi(x) = C f(x)
$$
to be real, because if it is complex, you can always do a rotation in the complex plane such that ##C' = C e^{i \delta}## is real.

But you also have the ##\lambda## in there and, unless told otherwise, you can't assume that it is real. You should appraoch the problem without restricting ##C## or ##\lambda## to be real, and see what you get.
 
5
0
use Maple to do the follow step:

assume(a>0)

int(C*(2*a^2*x^2+B)*exp(-(a^2*x^2)/2),x=-infinity..infinity)

where B is your λ. then we can get the result is
(B+2)*C*(2*Pi)^0.5/a

I hope this can help you
 

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