Quantum Mechanics is one of the most powerful and versatile areas of Physics, it allows to model intricate and complex systems which are usually difficult to describe with classical physics. The associated wavefunction can be used to predict the behavior of these systems, however representing these wavefunctions for digital computation can be challenging due to the high-dimensional nature of space-time, so describing these wavefunctions using classical mathematical notation has proven to be inefficient.
In this article we will explore how to use TensorFlow and its numerical annotation capabilities to create a continuous annotation of quantum mechanical wavefunctions that can be quickly and effortlessly parsed and used to perform calculations. This will offer a more efficient way to store a collection of quantum wavefunctions, making calculations faster and more accurate and allowing scientists to effectively explore complex quantum systems.
TensorFlow allows us to define a model of the quantum mechanical wavefunction as a tensor and to program in this notation to simplify the computational process. This way all operations involving the quantum wavefunction can be expressed as tensor operations and represented as a single computational graph.
To demonstrate this approach we will consider a one-dimensional quantum system defined using hyperparameters such as a Planck's constant, the system's velocity and its position. We will create a model of the quantum wavefunction with these parameters and use it to find the probability density of the particle's location.
The first step is to define the model using TensorFlow. We need to define our hyperparameters, like the Planck's constant h and the velocity v, and we can also set the number of samples we want to take for our quantum wavefunction. To simplify the notation we will define the quantum wavefunction using the following equation:
wave_function = tf.exp(-h * (v ** 2) * (position ** 2) / 2.0)
The next step is to create a TensorFlow graph that expresses the continuous annotation of the wavefunction. To do this, we will define a set of equations that will update the value of our wavefunction at each point in space-time. For this, we need to define the derivative of the wavefunction, which can be calculated as:
dfdx = tf.gradients(wave_function, position)[0]
We then need to calculate the gradient of the derivatives for the following equation:
ddfdx = tf.gradients(dfdx, position)[0]
Using these derivatives of the wavefunction we can then write our continuous annotation program. We can start by defining a placeholder variable in TensorFlow that will allow us to feed in data values at each point. We will then define a loop to iteratively update each of the variables in the quantum wavefunction equation:
for i in range(num_samples): # Update the derivative dfdx = tf.gradients(wave_function, position)[0] # Update the second derivative ddfdx = tf.gradients(dfdx, position)[0] # Calculate the change in position delta_x = 0.0001 * ddfdx # Calulate the new position new_x = position + delta_x # Update the wavefunction for the new position wave_function = tf.exp(-h * (v ** 2) * (new_x ** 2) / 2.0)
Using this approach we can create a continuous annotation program in TensorFlow to simulate the quantum system in one dimension. Once the program is written we can feed in values of the hyperparameters and collect the results of the computation. Here we have used one-dimensional space-time, however this approach can be extended to higher dimensions as well.
Using TensorFlow to program a continuous annotation of quantum mechanical wavefunctions allows scientists to quickly and easily explore complex quantum system while making calculations more efficient and accurate.