What is the Constant λ in the Wave Equation for Normalising the Wavefunction?

In summary, the homework statement is to find the constant λ in the wave equation. The attempt at a solution states that the wave equation just needs to be normalised. However, this is incorrect. The next step is also incorrect. The problem is to allow for ##C## and ##\lambda## to be real. If they are real, then \psi^{*} is the same as \psi.
  • #1
ParoxysmX
21
0

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]

Homework Equations



Some standard integrals I guess

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?
 
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  • #2
ParoxysmX said:
[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.
 
  • #3
If the function doesn't contain any complex exponentials, then [itex]\psi^{*}[/itex] is the same as [itex]\psi[/itex], isn't it?
 
  • #4
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
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
 

FAQ: What is the Constant λ in the Wave Equation for Normalising the Wavefunction?

1. What is wavefunction normalisation?

Wavefunction normalisation is a mathematical concept in quantum mechanics that ensures the total probability of finding a particle in any given location is equal to 1. It is a normalization constant applied to the wavefunction equation to account for all possible states of a system.

2. Why is wavefunction normalisation important?

Wavefunction normalisation is important because it ensures that the wavefunction accurately represents the probability of finding a particle in a particular location. Without normalisation, the total probability may exceed 1, which would violate the principles of quantum mechanics.

3. How is wavefunction normalisation calculated?

The wavefunction normalisation constant is calculated by taking the square root of the integral of the absolute value squared of the wavefunction over all space. This integral is known as the normalization integral and is denoted by the symbol ∫|ψ(x)|^2dx.

4. What is the physical significance of the wavefunction normalisation constant?

The wavefunction normalisation constant has no physical significance on its own. It is simply a mathematical tool used to ensure the wavefunction accurately represents the probability of finding a particle in a specific location. However, the square of the wavefunction, known as the probability density function, does have physical significance as it represents the probability of finding a particle at a given location.

5. Can a wavefunction be normalised to any value?

No, a wavefunction can only be normalised to a value of 1. This is because the total probability of finding a particle in any possible location must be equal to 1. If the normalisation constant is any other value, the wavefunction will not accurately represent the probability of finding the particle in a specific location.

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