Return to Question

The most attractive description of the double slit experiment for me is that in Beltrametti and Cassinelli's book.$^{[1]}$ The essence of their description is the following.

Beltrametti-Cassinelli description of the double-slit experiment - They somehow treat the $x$ position classically and suppose that the time of the collision to $S_2$ is independent of the position of the collision. They also adopt a reference frame in which the particle has no velocity along the x-axis. So the state space for the one-dimensional configuration space (i.e., along the y-axis) is $\mathfrak L^2(\mathbb R)$.

They take the state of the particle in the absence of the screens as a density operator $D$ and calculate the density operator $D'$ in the presence of screen $S_1$ as the density operator of the conditional probability in state $D$ of finding the particle in the interval $E$ at time instant $\tau$ with respect to the condition of finding it in $F_1\cup F_2$ at time instant $0$, that is, they take $$p_{D'}(\pi_E)=p_D(\pi_{E}(\tau)|\pi_{F_1\cup F_2}(0))\tag 1$$ where $\pi_E(\tau)$ is the projection belonging to the statement "at time instant $\tau$, the particle is located in $E$" and $\pi_{F_1\cup F_2}$ to "at time instant $0$, the particle is located in $F_1\cup F_2$". They use the Heisenberg picture, so that $$\pi(t)=U_t^{-1}\pi(0)U_t\tag 2$$ with the evolution operator $U_t$, and the state $D$ is independent of time. The conditional probability is defined as $$p_D(A|B)=\frac{\mathrm{tr}(DBAB)}{\mathrm{tr}(DB)}\tag 3,$$ that is, $$D'=\frac{BDB}{\mathrm{tr}(DB)}\tag {3'}.$$ At a point they take $D$ a pure state, that is, $D=D^\psi:=\psi\otimes\psi$ with a $\psi\in\mathcal H$ having unit norm). The story ends with $$p_D(\pi_E(\tau)|\pi_{F_1\cup F_2}(0)) = \int_E U_r(C_1\psi_1(y)+C_2\psi_2(y))|^2 dy \tag 4$$ where $\displaystyle \psi_i=\frac{\chi_{F_i}\psi}{\|\chi_{F_i}\psi\|}$ and $\displaystyle C_i=\frac{\|\chi_{F_i}\psi\|}{\|\chi_{F_1\cup F_2\|}}$ and $\chi_F$ is the characteristic function of the set $F$.

Question - The Aharonov-Bohm experiment is a slightly modified version of the double slit. Is this Beltrametti-Cassinelli kind of description applicable to the Aharonov-Bohm experiment too? Has this ever been attempted? If yes and not, then why not? Wouldn't be it more simple than working with path integrals?

$^{[1]}$ Enrico G. Beltrametti and Gianni Cassinelli - The Logic of Quantum Mechanics, Addison-Wesley, 1981, pp. 283 - 285

The most attractive description of the double slit experiment for me is that in Beltrametti and Cassinelli's book.$^{[1]}$ The essence of their description is the following.

Beltrametti-Cassinelli description of the double-slit experiment - They somehow treat the $x$ position classically and suppose that the time of the collision to $S_2$ is independent of the position of the collision. They also adopt a reference frame in which the particle has no velocity along the x-axis. So the state space for the one-dimensional configuration space (i.e., along the y-axis) is $\mathfrak L^2(\mathbb R)$.

They take the state of the particle in the absence of the screens as a density operator $D$ and calculate the density operator $D'$ in the presence of screen $S_1$ as the density operator of the conditional probability in state $D$ of finding the particle in the interval $E$ at time instant $\tau$ with respect to the condition of finding it in $F_1\cup F_2$ at time instant $0$, that is, they take $$p_{D'}(\pi_E)=p_D(\pi_{E}(\tau)|\pi_{F_1\cup F_2}(0))\tag 1$$ where $\pi_E(\tau)$ is the projection belonging to the statement "at time instant $\tau$, the particle is located in $E$" and $\pi_{F_1\cup F_2}$ to "at time instant $0$, the particle is located in $F_1\cup F_2$". They use the Heisenberg picture, so that $$\pi(t)=U_t^{-1}\pi(0)U_t\tag 2$$ with the evolution operator $U_t$, and the state $D$ is independent of time. The conditional probability is defined as $$p_D(A|B)=\frac{\mathrm{tr}(DBAB)}{\mathrm{tr}(DB)}\tag 3,$$ that is, $$D'=\frac{BDB}{\mathrm{tr}(DB)}\tag {3'}.$$ At a point they take $D$ a pure state, that is, $D=D^\psi:=\psi\otimes\psi$ with a $\psi\in\mathcal H$ having unit norm). The story ends with $$p_D(\pi_E(\tau)|\pi_{F_1\cup F_2}(0)) = \int_E |U_r(C_1\psi_1(y)+C_2\psi_2(y))|^2 dy \tag 4$$ where $\displaystyle \psi_i=\frac{\chi_{F_i}\psi}{\|\chi_{F_i}\psi\|}$ and $\displaystyle C_i=\frac{\|\chi_{F_i}\psi\|}{\|\chi_{F_1\cup F_2\|}}$ and $\chi_F$ is the characteristic function of the set $F$.

Question - The Aharonov-Bohm experiment is a slightly modified version of the double slit. Is this Beltrametti-Cassinelli kind of description applicable to the Aharonov-Bohm experiment too? Has this ever been attempted? If yes and not, then why not? Wouldn't be it more simple than working with path integrals?

$^{[1]}$ Enrico G. Beltrametti and Gianni Cassinelli - The Logic of Quantum Mechanics, Addison-Wesley, 1981, pp. 283 - 285

The most attractive description of the double slit experiment for me is that in Beltrametti and Cassinelli's book.$^{[1]}$ The essence of their description is the following.

Beltrametti-Cassinelli description of the double-slit experiment - They somehow treat the $x$ position classically and suppose that the time of the collision to $S_2$ is independent of the position of the collision. They also adopt a reference frame in which the particle has no velocity along the x-axis. So the state space for the one-dimensional configuration space (i.e., along the y-axis) is $\mathfrak L^2(\mathbb R)$.

They take the state of the particle in the absence of the screens as a density operator $D$ and calculate the density operator $D'$ in the presence of screen $S_1$ as the density operator of the conditional probability in state $D$ of finding the particle in the interval $E$ at time instant $\tau$ with respect to the condition of finding it in $F_1\cup F_2$ at time instant $0$, that is, they take $$p_{D'}(\pi_E)=p_D(\pi_{E}(\tau)|\pi_{F_1\cup F_2}(0))\tag 1$$ where $\pi_E(\tau)$ is the projection belonging to the statement "at time instant $\tau$, the particle is located in $E$" and $\pi_{F_1\cup F_2}$ to "at time instant $0$, the particle is located in $F_1\cup F_2$". They use the Heisenberg picture, so that $$\pi(t)=U_t^{-1}\pi(0)U_t\tag 2$$ with the evolution operator $U_t$, and the state $D$ is independent of time. The conditional probability is defined as $$p_D(A|B)=\frac{\mathrm{tr}(DBAB)}{\mathrm{tr}(DB)}\tag 3$$ At a point they take $D$ a pure state, that is, $D=D^\psi:=\psi\otimes\psi$ with a $\psi\in\mathcal H$ having unit norm). The story ends with $$p_D(\pi_E(\tau)|\pi_{F_1\cup F_2}(0)) = \int_E U_r(C_1\psi_1(y)+C_2\psi_2(y))|^2 dy \tag 4$$ where $\displaystyle \psi_i=\frac{\chi_{F_i}\psi}{\|\chi_{F_i}\psi\|}$ and $\displaystyle C_i=\frac{\|\chi_{F_i}\psi\|}{\|\chi_{F_1\cup F_2\|}}$ and $\chi_F$ is the characteristic function of the set $F$.

Question - The Aharonov-Bohm experiment is a slightly modified version of the double slit. Is this Beltrametti-Cassinelli kind of description applicable to the Aharonov-Bohm experiment too? Has this ever been attempted? If yes and not, then why not? Wouldn't be it more simple than working with path integrals?

$^{[1]}$ Enrico G. Beltrametti and Gianni Cassinelli - The Logic of Quantum Mechanics, Addison-Wesley, 1981, pp. 283 - 285

The most attractive description of the double slit experiment for me is that in Beltrametti and Cassinelli's book.$^{[1]}$ The essence of their description is the following.

Beltrametti-Cassinelli description of the double-slit experiment - They somehow treat the $x$ position classically and suppose that the time of the collision to $S_2$ is independent of the position of the collision. They also adopt a reference frame in which the particle has no velocity along the x-axis. So the state space for the one-dimensional configuration space (i.e., along the y-axis) is $\mathfrak L^2(\mathbb R)$.

They take the state of the particle in the absence of the screens as a density operator $D$ and calculate the density operator $D'$ in the presence of screen $S_1$ as the density operator of the conditional probability in state $D$ of finding the particle in the interval $E$ at time instant $\tau$ with respect to the condition of finding it in $F_1\cup F_2$ at time instant $0$, that is, they take $$p_{D'}(\pi_E)=p_D(\pi_{E}(\tau)|\pi_{F_1\cup F_2}(0))\tag 1$$ where $\pi_E(\tau)$ is the projection belonging to the statement "at time instant $\tau$, the particle is located in $E$" and $\pi_{F_1\cup F_2}$ to "at time instant $0$, the particle is located in $F_1\cup F_2$". They use the Heisenberg picture, so that $$\pi(t)=U_t^{-1}\pi(0)U_t\tag 2$$ with the evolution operator $U_t$, and the state $D$ is independent of time. The conditional probability is defined as $$p_D(A|B)=\frac{\mathrm{tr}(DBAB)}{\mathrm{tr}(DB)}\tag 3,$$ that is, $$D'=\frac{BDB}{\mathrm{tr}(DB)}\tag {3'}.$$ At a point they take $D$ a pure state, that is, $D=D^\psi:=\psi\otimes\psi$ with a $\psi\in\mathcal H$ having unit norm). The story ends with $$p_D(\pi_E(\tau)|\pi_{F_1\cup F_2}(0)) = \int_E U_r(C_1\psi_1(y)+C_2\psi_2(y))|^2 dy \tag 4$$ where $\displaystyle \psi_i=\frac{\chi_{F_i}\psi}{\|\chi_{F_i}\psi\|}$ and $\displaystyle C_i=\frac{\|\chi_{F_i}\psi\|}{\|\chi_{F_1\cup F_2\|}}$ and $\chi_F$ is the characteristic function of the set $F$.

Question - The Aharonov-Bohm experiment is a slightly modified version of the double slit. Is this Beltrametti-Cassinelli kind of description applicable to the Aharonov-Bohm experiment too? Has this ever been attempted? If yes and not, then why not? Wouldn't be it more simple than working with path integrals?

$^{[1]}$ Enrico G. Beltrametti and Gianni Cassinelli - The Logic of Quantum Mechanics, Addison-Wesley, 1981, pp. 283 - 285

The most attractive description of the double slit experiment for me is that in Beltrametti and Cassinelli's book.$^{[1]}$ The essence of their description is the following.

Beltrametti-Cassinelli description of the double-slit experiment - They somehow treat the $x$ position classically and suppose that the time of the collision to $S_2$ is independent of the position of the collision. They also adopt a reference frame in which the particle has no velocity along the x-axis. So the state space for the one-dimensional configuration space (i.e., along the y-axis) is $\mathfrak L^2(\mathbb R)$.

They take the state of the particle as a density operator $D$ and calculate the probability of finding the particle in the interval $E$ as the conditional probability $$p(\pi_{E}(\tau)|\pi_{F_1\cup F_2}(0))\tag 1$$ where $\pi_E(\tau)$ is the projection belonging to the statement "the particle hits the screen $S_2$ in $E$ at time instant $\tau$", and $\pi_{F_1\cup F_2}$ to "the particle is located in $F_1\cup F_2$ at time instant $0$". They use the Heisenberg picture, so that $$\pi(t)=U_t^{-1}\pi(0)U_t\tag 2$$ with the evolution operator $U_t$, and the state $D$ is independent of time. The conditional probability is defined as $$p(A|B)=\frac{\mathrm{tr}(DBAB)}{\mathrm{tr}(DB)}\tag 3$$ At a point they take the state a pure state, that is, $D=D^\psi:=\psi\otimes\psi$ with a $\psi\in\mathcal H$ having unit norm). The story ends with $$p(\pi_E(\tau)|\pi_{F_1\cup F_2}(0)) = \int_E U_r(C_1\psi_1(y)+C_2\psi_2(y))|^2 dy \tag 4$$ where $\displaystyle \psi_i=\frac{\chi_{F_i}\psi}{\|\chi_{F_i}\psi\|}$ and $\displaystyle C_i=\frac{\|\chi_{F_i}\psi\|}{\|\chi_{F_1\cup F_2\|}}$ and $\chi_F$ is the characteristic function of the set $F$.

Question - The Aharonov-Bohm experiment is a slightly modified version of the double slit. Is this Beltrametti-Cassinelli kind of description applicable to the Aharonov-Bohm experiment too? Has this ever been attempted? If yes and not, then why not? Wouldn't be it more simple than working with path integrals?

$^{[1]}$ Enrico G. Beltrametti and Gianni Cassinelli - The Logic of Quantum Mechanics, Addison-Wesley, 1981, pp. 283 - 285

The most attractive description of the double slit experiment for me is that in Beltrametti and Cassinelli's book.$^{[1]}$ The essence of their description is the following.

Beltrametti-Cassinelli description of the double-slit experiment - They somehow treat the $x$ position classically and suppose that the time of the collision to $S_2$ is independent of the position of the collision. They also adopt a reference frame in which the particle has no velocity along the x-axis. So the state space for the one-dimensional configuration space (i.e., along the y-axis) is $\mathfrak L^2(\mathbb R)$.

They take the state of the particle in the absence of the screens as a density operator $D$ and calculate the density operator $D'$ in the presence of screen $S_1$ as the density operator of the conditional probability in state $D$ of finding the particle in the interval $E$ at time instant $\tau$ with respect to the condition of finding it in $F_1\cup F_2$ at time instant $0$, that is, they take $$p_{D'}(\pi_E)=p_D(\pi_{E}(\tau)|\pi_{F_1\cup F_2}(0))\tag 1$$ where $\pi_E(\tau)$ is the projection belonging to the statement "at time instant $\tau$, the particle is located in $E$" and $\pi_{F_1\cup F_2}$ to "at time instant $0$, the particle is located in $F_1\cup F_2$". They use the Heisenberg picture, so that $$\pi(t)=U_t^{-1}\pi(0)U_t\tag 2$$ with the evolution operator $U_t$, and the state $D$ is independent of time. The conditional probability is defined as $$p_D(A|B)=\frac{\mathrm{tr}(DBAB)}{\mathrm{tr}(DB)}\tag 3$$ At a point they take $D$ a pure state, that is, $D=D^\psi:=\psi\otimes\psi$ with a $\psi\in\mathcal H$ having unit norm). The story ends with $$p_D(\pi_E(\tau)|\pi_{F_1\cup F_2}(0)) = \int_E U_r(C_1\psi_1(y)+C_2\psi_2(y))|^2 dy \tag 4$$ where $\displaystyle \psi_i=\frac{\chi_{F_i}\psi}{\|\chi_{F_i}\psi\|}$ and $\displaystyle C_i=\frac{\|\chi_{F_i}\psi\|}{\|\chi_{F_1\cup F_2\|}}$ and $\chi_F$ is the characteristic function of the set $F$.

Question - The Aharonov-Bohm experiment is a slightly modified version of the double slit. Is this Beltrametti-Cassinelli kind of description applicable to the Aharonov-Bohm experiment too? Has this ever been attempted? If yes and not, then why not? Wouldn't be it more simple than working with path integrals?

$^{[1]}$ Enrico G. Beltrametti and Gianni Cassinelli - The Logic of Quantum Mechanics, Addison-Wesley, 1981, pp. 283 - 285