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Chapter 1: Problem 16

Show that if the Schrödinger equation had the form of a true wave equation, then the integrated probability would be time dependent. Hint. A wave equation has \(\kappa \partial^{2} / \partial t^{2}\) in place of it a chere \(\kappa\) is a constant with the appropriate dimensions (what are they?). Solve the time component of the separable equation and investigate the behaviour of \(\int \Psi^{*} \Psi d \tau\)

Short Answer

Expert verified
Replacing the first order time derivative in the Schrödinger equation with the second order time derivative causes the total probability of finding the particle anywhere in space to become time dependent. This is inconsistent with quantum mechanics, illustrating that the Schrödinger equation isn't a regular wave equation.

Step by step solution

01

Replace the Time-Derivative in the Schrödinger equation

The standard Schrödinger equation is expressed as \[- i \hbar \frac{\partial}{\partial t} \Psi = H \Psi\]Replace the first order time derivative \(\frac{\partial}{\partial t}\) with the second order time derivative \(\frac{\partial^{2}}{\partial t^{2}}\). The modified Schrödinger equation becomes \[\kappa \frac{\partial^{2}}{\partial t^{2}} \Psi = H \Psi\] where \(\kappa\) is a constant factor with dimensions \( \frac{ [Time]^{2} }{ [Energy] } \). This is a direct analog to the common wave equation.
02

Solve the Modified Schrödinger Equation

Solve for \(\Psi\) in the modified equation. The general solution of a second-order derivative equation involves sines and cosines or complex exponentials.
03

Evaluating the total probability

Next, consider the total probability of finding the particle anywhere in space. This is given by the integral of the absolute value squared of the wave function over all space. \[\int \Psi^{*} \Psi d\tau\] Substitute \(\Psi\) from the solved equation into this formula, then compute the integral and observe whether it is time-dependent.

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Most popular questions from this chapter

Write the time-independent Schrödinger equations for (a) the hydrogen atom, (b) the helium atom, (c) the hydrogen molecule, (d) a free particle, (e) a particle subjected to a constant, uniform force. Hint. Identify the appropriate potential energy terms and express them as operators in the position representation.

Show that (a) \([A, B]=-[B, A],\) (b) \(\left[A^{m}, A^{n}\right]=0\) for all \(m, n,\) (c) \(\left[A^{2}, B\right]=A[A, B]+[A, B] A\) (d) \([A,[B, C]]+[B,[C, A]]+[C,[A, B]]=0\)

The operator \(e^{A}\) has a meaning if it is expanded as a power scrics: \(\mathrm{e}^{A}=\Sigma_{n}(1 / n !) A^{n} .\) Show that if \(|a\rangle\) is an eigenstate of \(A\) with eigenvalue \(a,\) then it is also an eigenstate of \(\mathrm{e}^{A} .\) Find the latter's eigenvalue.

Evaluate the commutators (a) \(\left[H, p_{x}\right]\) and (b) \([H, x]\) where \(H=p_{x}^{2} / 2 m+V(x) .\) Choose (i) \(V(x)=V,\) a constant, (ii) \(V(x)=\frac{1}{2} k_{t} x^{2},\) (iii) \(V(x) \rightarrow V(r)=e^{2} / 4 \pi \varepsilon_{0} r .\) Hint. For part (b), case (iii), use \(\left(\partial r^{-1} / \partial x\right)=-x / r^{3}\)

Evaluate the expectation values of the operators \(p_{x}\) and \(p_{x}^{2}\) for a particle with wavefunction \((2 / L)^{1 / 2} \sin (\pi x / L)\) in the range 0 to \(L\)
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