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Q9D.1 (Requires studying appendix QA.) The eigenfunction corresponding to a definite value $p_0$ of a quanton's x-momentum is the function $\Psi_{p_0}(x) = Ae^{ip_0x/\hbar} = A \cos(p_0x/\hbar) + iA \sin(p_0x/\hbar)$ (Q9.24) (a) Argue that the wavelengths of the cosine and sine parts of this wave (considered separately) are both $\lambda = h/p_0$, as specified by the de Broglie formula. (b) Assume that we determine a quanton's x-momentum and get the value $p_0$, collapsing the state to the momentum eigen-function given above. Argue that if we then determine the quanton's position, the probability of any particular outcome x is proportional to $|A|^2$, independent of x. Since all position outcomes from $-\infty$ to $+\infty$ are equally likely, the uncertainty in the quanton's position is therefore infinite when we are in a state of definite momentum, consistent with the Heisenberg Uncertainty Principle given by equation Q9.17. (c) Now suppose we determine a quanton's x-position and get the value $x_0$, collapsing the state to a position eigenfunction (a spike). Show that if we then determine the quanton's x-momentum, the probability of getting any particular mo-mentum value $p_0$ is proportional to $|A|^2$, independent of $p_0$. Since all momentum outcomes from $-\infty$ to $+\infty$ are equally likely, the uncertainty in the quanton's x-momentum is therefore infinite when we are in a state of definite position. This is again consistent with the Heisenberg Uncertainty Principle. (Hint: Note that $(\psi_p, x) = (x \psi_p) = [(x \psi_p)]^*)$.

          Q9D.1 (Requires studying appendix QA.) The eigenfunction corresponding to a definite value $p_0$ of a quanton's x-momentum is the function
$\Psi_{p_0}(x) = Ae^{ip_0x/\hbar} = A \cos(p_0x/\hbar) + iA \sin(p_0x/\hbar)$ (Q9.24)
(a) Argue that the wavelengths of the cosine and sine parts of this wave (considered separately) are both $\lambda = h/p_0$, as specified by the de Broglie formula.
(b) Assume that we determine a quanton's x-momentum and get the value $p_0$, collapsing the state to the momentum eigen-function given above. Argue that if we then determine the quanton's position, the probability of any particular outcome x is proportional to $|A|^2$, independent of x. Since all position outcomes from $-\infty$ to $+\infty$ are equally likely, the uncertainty in the quanton's position is therefore infinite when we are in a state of definite momentum, consistent with the Heisenberg Uncertainty Principle given by equation Q9.17.
(c) Now suppose we determine a quanton's x-position and get the value $x_0$, collapsing the state to a position eigenfunction (a spike). Show that if we then determine the quanton's x-momentum, the probability of getting any particular mo-mentum value $p_0$ is proportional to $|A|^2$, independent of $p_0$. Since all momentum outcomes from $-\infty$ to $+\infty$ are equally likely, the uncertainty in the quanton's x-momentum is therefore infinite when we are in a state of definite position. This is again consistent with the Heisenberg Uncertainty Principle. (Hint: Note that $(\psi_p, x) = (x \psi_p) = [(x \psi_p)]^*)$.

Q9D.1 (Requires studying appendix QA.) The eigenfunction corresponding to a definite value p0 of a quanton's x-momentum is the function
Ψp0(x) = Ae^ip0x/ħ = A cos(p0x/ħ) + iA sin(p0x/ħ) (Q9.24)
(a) Argue that the wavelengths of the cosine and sine parts of this wave (considered separately) are both λ = h/p0, as specified by the de Broglie formula.
(b) Assume that we determine a quanton's x-momentum and get the value p0, collapsing the state to the momentum eigen-function given above. Argue that if we then determine the quanton's position, the probability of any particular outcome x is proportional to |A|^2, independent of x. Since all position outcomes from -∞ to +∞ are equally likely, the uncertainty in the quanton's position is therefore infinite when we are in a state of definite momentum, consistent with the Heisenberg Uncertainty Principle given by equation Q9.17.
(c) Now suppose we determine a quanton's x-position and get the value x0, collapsing the state to a position eigenfunction (a spike). Show that if we then determine the quanton's x-momentum, the probability of getting any particular mo-mentum value p0 is proportional to |A|^2, independent of p0. Since all momentum outcomes from -∞ to +∞ are equally likely, the uncertainty in the quanton's x-momentum is therefore infinite when we are in a state of definite position. This is again consistent with the Heisenberg Uncertainty Principle. (Hint: Note that (, x) = (x ) = [(x )]^*).

Added by Justin M.

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