Principle of superposition and Uncertainty principle I think "the principle of superposition" and "the uncertainty principle" are two different and distinct principles used in quantum mechanics. Uncertainty principle is not the cause of Superposition principle and vice versa.
Uncertainty principle: It says that it is impossible to measure conjugate measurables such as position and momentum simultaneously. Classical mechanics says that "if one knows the positions and momenta of particles of a system then one can predict the future states of the system by using Newton's laws". This is classical "determinacy". Thus uncertainty principle negates the classical determinacy by saying that you "can not measure" the position and momentum of a system simultaneously; then how can you specify them as demanded by determinacy idea? So the uncertainty principle is a purely non classical principle.
Principle of superposition: Classical mechanics says that a particle can stay in a particular state in a particular instant of time. But experimental observations of quantum systems show that a system can remain in any of the many states those satisfy the Schrödinger equation. However there is a certain probability associated with each feasible state. Thus the general state of the system is a superposition of all the possible states weighted by certain probability. These probabilities depend on the way the system was prepared. It is like assigning amplitudes to different normal modes depending on where the string was plucked. I can prepare system with only up spin atoms by doing a SG experiment. Thus there is no uncertainty in superposition principle.
Principle of superposition is also a non-classical principle.
But my problem starts as follows:
Please refer to the book "No-nonsense quantum mechanics" by Jakob Schwichtenberg, page number 49 and 50. There, while trying to invoke and explain the Superposition principle, the author clearly mentions that the idea behind invoking superposition principle was to incorporate quantum uncertainty. I am finding it difficult to accept.
Yes, I agree that the superposition principle invokes an idea of "Probablity" that is inherent in quantum mechanics. But this probability has nothing to do with quantum uncertainty. Quantum uncertainty is inherent in the very act of measurement and is concerned with conjugate measurables only. But probability of a particular state is related to Hamiltonian of the system. Hence the Principle of superposition and the Uncertainty principle are two distinct and different principles. One is not the cause of the other.
Is uncertainty principle the cause of invoking superposition principle?
Or are they different principles?
 A: Boring Stage-Setting/Caveats
I'd try to give a very hand-wavy intuition as to why the two principles, while distinct, are very closely related -- so much so that I can't imagine the existence of one without some form of the other -- and vice versa.
To streamline the discussion, I'd replace the uncertainty principle with the principle that not all observables are compatible. It can be shown easily (see Robertson-Schrodinger uncertainty relations) that the uncertainty principle (that you cannot reduce the conjoint uncertainty in the measurement of conjugate observables below a certain value) stems directly from the fact that certain observables (namely, the conjugate observables in the original formulation of Heisenberg) do not commute with each other.
Finally, I don't have the book you mentioned at the moment, so I'd provide an answer simply based on the information available in your post.
The Core Argument
Uncertainty $\to$ Superposition

*

*The existence of incompatible observables means that there exist states of a system that have a well-defined value of one observable but they do not have a well-defined value of another observable. Notice that this is the core non-classical aspect of quantum mechanics, nothing like this can be said for a classical system. A classical system always has well-defined values for all canonical variables that one can define.

*Now, consider a system that has a well-defined value for an observable $A$ but does not have a well-defined value for an observable $B$. However, if you choose to measure $B$ on this system, you'd get some result. Of course, since the system does not have a well-defined value for $B$, the result you get would vary each time you do the experiment and would only obey a probability distribution.

*Thus, you need a formalism that describes such a state of the system as some form of a superposition
of states with different values of $B$. Of course, that this superposition is a linear superposition is not apriori but is deduced from experimental results. So, we have argued that the existence of incompatible observables almost forces us to adopt a formalism that involves some form of a superposition of states with definite values of an observable.

Superposition $\to$ Uncertainty

*

*Similarly, you can also argue the other way around. A simple argument is that if all observables were compatible, i.e., can be assigned definite values simultaneously, then the formalism simply doesn't need a superposition of states. Thus, in order to have a meaningful concept of superposition of states, one would need to have incompatible observables.

*Finally, I would argue the previous point in a different language which requires a slight familiarity with quantum mechanics: Imagine that you had a superposition of states. Now, if all the observables commuted then you could just measure the state for any of the observables and the resultant state would not be a non-trivial superposition w.r.t. the spectrum of any of the observables. Nor would it evolve to such a non-trivial superposition, by stipulation, it would also be a state with definite energy.


Warning
It should be noted that this is just a hand-wavy way of showing that the existence of a concept of superposition in a formalism that has incompatible observables (and vice-versa) is well-motivated on physical grounds. In no way should this be taken to understand that incompatibility of observables or superposition of states is implied by the other (despite the $\to$ sign in my headings) in any strict sense.
A: I think you are confusing two concepts - quantum uncertainty/superposition and Heisenberg's uncertainty principle. They are different.
Heisenberg's uncertainty principle arises from the existence of pairs of operators with non-zero commutators. These pairs correspond to conjugate variables. However, it is perfectly possible to have a quantum system that only has one variable. For example, a single qubit has just one measurable variable, $\Phi$, which can either take the value $0$ or the value $1$. In this simple system, there are no pairs of conjugate variables because the only pair of variables is $(\Phi, \Phi)$ and, of course, the commutator $[\Phi, \Phi ]$ is zero.
However, quantum uncertainty still applies to a single qubit, and the state of the qubit can be represented as the superposition of its two orthogonal eigenstates
$$|\phi \rangle = \alpha |0\rangle + \beta |1\rangle$$
So quantum uncertainty and the superposition principle still applies in a system that has no pairs of conjugate variables.
